Short answer
Most Buddhists bow to the Buddha, but only some ask him or a bodhisattva (a being vowed to save others) for help. It depends on the school. Theravada Buddhists take refuge in the Buddha and chant verses for protection. Yet the Pali texts have him say that the Buddhas only show the way. Many Pure Land and other Mahayana Buddhists do call on Amitabha or Avalokitesvara to rescue them. Islam gives one answer for all. The Qur'an says to invoke no one along with Allah (72:18). Even the Prophet (ﷺ) is told to say that he only invokes his Lord and has no power to bring anyone to right conduct (72:20-21). Allah (﷾) hears those who call on Him (40:60), and no one can plead with Him except after His leave (2:255).
Full explanation
Two branches, and who calls on whom
Buddhism has two great branches. Theravada, found mostly in South and Southeast Asia, follows the Pali texts. Mahayana, found mostly in East Asia, adds further sutras and honours many bodhisattvas. Zen and Pure Land are Mahayana schools. A Theravada Buddhist becomes a Buddhist by going for refuge to the Buddha, the Teaching and the Sangha, the community of his disciples. Many Pure Land and other Mahayana Buddhists also call on Amitabha or Avalokitesvara by name. Many Zen practitioners pray to no one and sit in meditation. This page says which school each practice belongs to, and it does not describe one school's practice as another's.
“I take refuge in the Buddha, I take refuge in the Teaching, I take refuge in the Saṅgha.”
Theravada: refuge, bowing and verses for protection
In the Pali texts the Buddha's followers go for refuge to him, to his teaching and to the Saṅgha. The Dhammapada says that this refuge leads the one who goes to it to see the four noble truths. It also says that to venerate the Buddha or his disciple brings great merit. Theravada Buddhists also chant the Ratana Sutta as a protective chant (paritta). Its refrain, "by this truth, may you be well!", is said to the beings gathered. At its close the chanter says "We bow to the Buddha! May you be safe!" to those same beings. The texts quoted here present honour for a teacher who has passed beyond suffering. They do not present a Buddha who hears and answers a caller.
“I take refuge in the Buddha, I take refuge in the Teaching, I take refuge in the Saṅgha.”
“One gone for refuge to the Buddha, to his teaching and to the Saṅgha, sees the four noble truths with right understanding: suffering, suffering's origin, suffering's transcendence, and the noble eightfold path that leads to the stilling of suffering. Such refuge is a sanctuary, it is the supreme refuge. By going to that refuge, you're released from all suffering.”
“When a person venerates the worthy—the Buddha or his disciple, who have transcended proliferation, and have left behind grief and lamentation, quenched, fearing nothing from any quarter—the merit of one venerating such as these, cannot be calculated by anyone, saying it is just this much.”
“There's no wealth in this life or the next, no sublime gem in the heavens, that equals the Realized One. This sublime gem is in the Buddha: by this truth, may you be well!”
“Whatever beings have gathered here, on the ground or in the sky: the Realized One is honored by gods and humans! We bow to the Buddha! May you be safe!”
What the Pali Buddha says of himself
In the Dhammapada the Buddha tells his hearers that they must do the work themselves, and that the Buddhas only show the way. Another verse says that one is the lord of oneself. Near the end of his life, in the Long Discourses, he tells Ananda that the monks are to live as their own island and refuge. The teaching is to be their island and refuge too, and no other. In these Pali texts the Buddha points away from himself, toward effort and the teaching.
“You yourselves must do the work, the Realized Ones just show the way.”
“One is indeed the lord of oneself, for who else would be one's lord? By means of a well-tamed self, one gains a lord that's rare indeed.”
“So Ānanda, live as your own island, your own refuge, with no other refuge. Let the teaching be your island and your refuge, with no other refuge.”
Mahayana: Avalokitesvara and Amitabha
The Lotus Sutra is a Mahayana sutra, not a Pali text. In it the Buddha says that those who hear the name of the bodhisattva Avalokitesvara, or implore him, are released from troubles. The troubles named include fire and flood. The Larger Sukhavativyuha, also Mahayana, is one of the main texts of Pure Land Buddhism. In its prayer 19, whose closing part matches the well-known 18th vow of the Chinese text, Amitabha makes a vow before he becomes a Buddha. Those who turn their thought to be born in his land, even with ten thoughts of it, shall be born there. The vow bars those who have done the five gravest wrongs or who abuse the good Law. These are the texts on which the calling on a name rests.
“All the hundred thousands of myriads of kotis of creatures, young man of good family, who in this world are suffering troubles will, if they hear the name of the Bodhisattva Mahasattva Avalokitesvara, be released from that mass of troubles. Those who shall keep the name of this Bodhisattva Mahasattva Avalokitesvara, young man of good family, will, if they fall into a great mass of fire, be delivered therefrom by virtue of the lustre of the Bodhisattva Mahasattva. In case, young man of good family, creatures, carried off by the current of rivers, should implore the Bodhisattva Mahasattva Avalokitesvara, all rivers will afford them a ford.”
“O Bhagavat, if those beings who in immeasurable and innumerable Buddha countries, after they have heard my name, when I shall have obtained Bodhi, should direct their thought to be born in that Buddha country of mine, and should for that purpose bring their stock of merit to maturity, if these should not be born in that Buddha country, even those who have only ten times repeated the thought (of that Buddha country), barring always those beings who have committed the (five) Anantarya sins, and who have caused an obstruction and abuse of the good Law, then may I not obtain the highest perfect knowledge.”
A teacher who points away from himself, and to whom
The Qur'an has the Prophet (ﷺ) say that he does no more than invoke his Lord and joins no one with Him. He says too that it is not in his power to cause harm, or to bring anyone to right conduct. He is, he says, only a man like others to whom the inspiration has come that God (﷾) is one. So the Buddha of the Pali texts and the Prophet of the Qur'an both point away from themselves. Islam says to whom: to the One who made us and who alone can answer.
“Say: "I do no more than invoke my Lord, and I join not with Him any (false god)." Say: "It is not in my power to cause you harm, or to bring you to right conduct."”
“Say: "I am but a man like yourselves, (but) the inspiration has come to me, that your Allah is one Allah: whoever expects to meet his Lord, let him work righteousness, and, in the worship of his Lord, admit no one as partner.”
“You yourselves must do the work, the Realized Ones just show the way.”
The One who hears, and who is near
The Qur'an records Allah (﷾) saying, "Call on Me; I will answer your (Prayer)". It says that He is indeed close to His servants and listens to every suppliant who calls. It says that to Him belongs all dominion. Those invoked besides Him have not the least power, will not listen to the call and cannot answer. The Muslim therefore asks Allah (﷾) alone. The first chapter of the Qur'an, recited in every prayer, says "Thee do we worship, and Thine aid we seek". Prayer, belief and character together make one way of life, and the believer's asking is part of it.
“And your Lord says: "Call on Me; I will answer your (Prayer): but those who are too arrogant to serve Me will surely find themselves in Hell - in humiliation!"”
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
“He merges Night into Day, and he merges Day into Night, and he has subjected the sun and the moon (to his Law): each one runs its course for a term appointed. Such is Allah your Lord: to Him belongs all Dominion. And those whom ye invoke besides Him have not the least power.”
“If ye invoke them, they will not listen to your call, and if they were to listen, they cannot answer your (prayer). On the Day of Judgment they will reject your "Partnership". and none, (O man!) can tell thee (the Truth) like the One Who is acquainted with all things.”
“Thee do we worship, and Thine aid we seek.”
Invoke no one along with Allah (﷾)
The Qur'an says that the places of worship are for Allah (﷾), so no one is to be invoked along with Him. This is the rule for every reader, Muslim or not. It does not forbid respect for a teacher. It forbids offering a person the calling on, the trust and the devotion that belong to Allah (﷾) alone.
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
Intercession: only by His leave
Islam does not deny that intercession exists. The Qur'an says who may plead with Allah (﷾): no one, except after His leave, and only for those whom He accepts. It says that even the angels' intercession avails nothing except after Allah (﷾) has given leave. So intercession is no power that anyone holds in his own right. No person is a way round Allah (﷾), to be asked in His place. On the Day the Prophet (ﷺ) will intercede, but only after Allah (﷾) gives leave and only for those He accepts. So the believer asks Allah (﷾) for that intercession, and asks no one in His place.
“Allah! There is no god but He,-the Living, the Self-subsisting, Eternal. No slumber can seize Him nor sleep. His are all things in the heavens and on earth. Who is there can intercede in His presence except as He permitteth? He knoweth what (appeareth to His creatures as) before or after or behind them. Nor shall they compass aught of His knowledge except as He willeth. His Throne doth extend over the heavens and the earth, and He feeleth no fatigue in guarding and preserving them for He is the Most High, the Supreme (in glory).”
“Verily your Lord is Allah, who created the heavens and the earth in six days, and is firmly established on the throne (of authority), regulating and governing all things. No intercessor (can plead with Him) except after His leave (hath been obtained). This is Allah your Lord; Him therefore serve ye: will ye not receive admonition?”
“On that Day shall no intercession avail except for those for whom permission has been granted by (Allah) Most Gracious and whose word is acceptable to Him.”
“How many-so-ever be the angels in the heavens, their intercession will avail nothing except after Allah has given leave for whom He pleases and that he is acceptable to Him.”
Abu Huraira reports that the Prophet (ﷺ) was asked who would be the happiest of people with his intercession on the Day of Resurrection. He told Abu Huraira that, seeing his eagerness for hadith, he had expected no one to ask him about it before him. Then he said it would be the one who said "there is no god but Allah" sincerely (khalisan), from his heart.
"We honour them, we do not worship them"
The Qur'an records a plea from those who take protectors other than Allah (﷾). They say: "We only serve them in order that they may bring us nearer to Allah". It answers that sincere devotion is due to Allah (﷾), and that Allah (﷾) will judge between those who differ. Elsewhere those who serve things besides Him say "These are our intercessors with Allah". The Qur'an asks whether they inform Allah (﷾) of something He does not know. The principle reaches every reader: no go-between is needed to call on Him, since Allah (﷾) is close and hears.
“Is it not to Allah that sincere devotion is due? But those who take for protectors other than Allah (say): "We only serve them in order that they may bring us nearer to Allah." Truly Allah will judge between them in that wherein they differ. But Allah guides not such as are false and ungrateful.”
“They serve, besides Allah, things that hurt them not nor profit them, and they say: "These are our intercessors with Allah." Say: "Do ye indeed inform Allah of something He knows not, in the heavens or on earth?- Glory to Him! and far is He above the partners they ascribe (to Him)!"”
“When My servants ask thee concerning Me, I am indeed close (to them): I listen to the prayer of every suppliant when he calleth on Me: Let them also, with a will, Listen to My call, and believe in Me: That they may walk in the right way.”
The One to call on
The Qur'an asks to be weighed with care. It was brought by Muhammad (ﷺ), who was sent to all mankind. Muslims invite every reader, Buddhists too, to read it and to call on the One it describes.
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“Say: "O men! I am sent unto you all, as the Messenger of Allah, to Whom belongeth the dominion of the heavens and the earth: there is no god but He: it is He That giveth both life and death. So believe in Allah and His Messenger, the Unlettered Prophet, who believeth in Allah and His words: follow him that (so) ye may be guided."”
Six replies answered
Khuddakapatha 1 — Basic Passages, passage 1 (The Three Refuges)
“I take refuge in the Buddha, I take refuge in the Teaching, I take refuge in the Saṅgha.”
Cited for
Theravada Buddhists only go for refuge, bow and chant for protection. That is respect for a teacher, and you have no quarrel with it.
The response
That is fair, and this page does not say Theravadins pray to the Buddha as to a god. The Pali texts present honour, and the Buddha himself says the Buddhas only show the way (Dhammapada 276). The question that remains is where a person turns in need. The Qur'an answers: Allah (﷾) says "Call on Me; I will answer your (Prayer)" (40:60), and He is close (2:186).
Where it lands — No claim is made against Theravada practice as the texts present it. The question is whom a person calls on in need.
Dhammapada 195-196 — The Buddhas, verses 195-196
“When a person venerates the worthy—the Buddha or his disciple, who have transcended proliferation, and have left behind grief and lamentation, quenched, fearing nothing from any quarter—the merit of one venerating such as these, cannot be calculated by anyone, saying it is just this much.”
Cited for
We honour the Buddha, we do not worship him. A bow is respect, and nobody is asking him for anything.
The response
Respect for a teacher is not what Islam forbids. The Qur'an forbids calling on anyone along with Allah (72:18). It also records the plea of those who take other protectors: "We only serve them in order that they may bring us nearer to Allah". It answers that sincere devotion is due to Allah (39:3). Where honour stays honour, there is nothing to answer. Where it turns to asking for rescue, the Qur'an's rule applies.
Where it lands — Honour is not the question. Calling on someone for help, and giving him the devotion that belongs to Allah (﷾), is.
Larger Sukhavativyuha (Description of Sukhavati) 8.19 — section 8, prayer 19, p. 15
“O Bhagavat, if those beings who in immeasurable and innumerable Buddha countries, after they have heard my name, when I shall have obtained Bodhi, should direct their thought to be born in that Buddha country of mine, and should for that purpose bring their stock of merit to maturity, if these should not be born in that Buddha country, even those who have only ten times repeated the thought (of that Buddha country), barring always those beings who have committed the (five) Anantarya sins, and who have caused an obstruction and abuse of the good Law, then may I not obtain the highest perfect knowledge.”
Cited for
We are saved by Amitabha's power, not our own (other-power). Saying his name is gratitude for a rescue already given, not a petition.
The response
Gratitude is still spoken to someone who hears. The vow, in the text that Pure Land rests on, promises rebirth in his land to those who turn to it, in countless worlds. It bars only those who have done the five gravest wrongs or abused the good Law. A being who hears every such caller and saves him does what the Qur'an gives to Allah (﷾) alone (40:60; 35:13-14). The Pali Buddha says the Buddhas only show the way (Dhammapada 276).
Where it lands — If Amitabha hears every caller and saves, he is given the power that Islam gives to Allah (﷾) alone.
Saddharma-pundarika (Lotus Sutra) 24 — chapter XXIV (The All-Sided One, the transformations of Avalokitesvara), p. 406
“All the hundred thousands of myriads of kotis of creatures, young man of good family, who in this world are suffering troubles will, if they hear the name of the Bodhisattva Mahasattva Avalokitesvara, be released from that mass of troubles. Those who shall keep the name of this Bodhisattva Mahasattva Avalokitesvara, young man of good family, will, if they fall into a great mass of fire, be delivered therefrom by virtue of the lustre of the Bodhisattva Mahasattva. In case, young man of good family, creatures, carried off by the current of rivers, should implore the Bodhisattva Mahasattva Avalokitesvara, all rivers will afford them a ford.”
Cited for
Avalokitesvara is the bodhisattva of compassion. Calling his name is how compassion reaches us, and it does not ask a person for anything.
The response
The Lotus Sutra says that creatures in fire or in flood who hear or implore his name are released. That is a promise of rescue to every caller. If the figure is a symbol, nothing hears the caller. If it is a being, it does what the Qur'an gives to Allah (﷾), who is close and answers (2:186; 40:60). The Qur'an says those invoked besides Him cannot answer (35:14).
Where it lands — A symbol cannot hear, and a being who hears every caller has the power of Allah (﷾) alone.
Digha Nikaya 16 — Long Discourses, discourse 16 (The Great Discourse on the Final Extinguishment), segment 2.26.1
“So Ānanda, live as your own island, your own refuge, with no other refuge. Let the teaching be your island and your refuge, with no other refuge.”
Cited for
Many Zen practitioners, and the Buddha himself in the Pali texts, say to rely on yourself. I pray to no one, so none of this applies to me.
The response
Then this page does not say that you pray, and Islam honours effort and a clean heart. The Qur'an adds a question that effort cannot answer: who made us, and who keeps us. It says the One who does is close, hears the one who calls, and is to be asked (2:186; 40:60).
Where it lands — Self-reliance is half of the road. The Qur'an adds the One who made us and answers.
Qur'an 2:255 — Surat al-Baqara (The Cow), verse 255
“Allah! There is no god but He,-the Living, the Self-subsisting, Eternal. No slumber can seize Him nor sleep. His are all things in the heavens and on earth. Who is there can intercede in His presence except as He permitteth? He knoweth what (appeareth to His creatures as) before or after or behind them. Nor shall they compass aught of His knowledge except as He willeth. His Throne doth extend over the heavens and the earth, and He feeleth no fatigue in guarding and preserving them for He is the Most High, the Supreme (in glory).”
Cited for
Muslims also ask the Prophet (ﷺ) to intercede with Allah and visit graves to ask for help. Islam does the same as the Buddhists you criticise.
The response
Islam allows visiting graves (Muslim 977a). It forbids calling on the dead for what only Allah (﷾) gives: "invoke not any one along with Allah" (72:18). All intercession belongs to Allah (39:44). The Prophet (ﷺ) will intercede on the Day, but only after Allah's (﷾) leave, for those He accepts (2:255; 20:109; Bukhari 99). So a Muslim asks Allah (﷾) for that intercession, and asks no one in His place. The page "Can a Muslim call on anyone but Allah for help?" takes it up.
Where it lands — Islam's rule is stated and quoted: invoke none with Allah (﷾), and intercession only by His leave.
Perspectives
Theravada Buddhist teaching (the Pali texts)
A Theravada Buddhist replies from the Pali texts. He goes for refuge to the Buddha, bows to him and chants verses of protection, as honour and gratitude. The Buddha himself says the Buddhas only show the way and that each hearer must be his own island and refuge.
Theravada, which follows the Pali texts
Theravada Buddhists go for refuge to the Buddha, bow to him and chant verses for protection. This is honour for a teacher, not prayer to a god, and the Pali Buddha says the Buddhas only show the way.
Rebuttal
Granted, and Islam does not say Theravadins pray to the Buddha as to a god. What is left is the need that prayer meets, and the Qur'an answers it: "Call on Me; I will answer your (Prayer)" (40:60).
Going for refuge leads to seeing the four noble truths, and to venerate the Buddha or his disciple brings great merit (Dhammapada 190-192; 195-196). So honouring the Buddha is a religious act that matters.
Rebuttal
Granted that the verses tie refuge to understanding and veneration to merit. Neither verse asks the Buddha to hear or act. Islam holds that merit, and refuge itself, are in Allah's (﷾) hands.
We honour the Buddha, we do not worship him. A bow is respect, as a pupil bows to a teacher.
Rebuttal
Respect for a teacher is not what the Qur'an forbids. It forbids calling on anyone along with Allah (72:18). It also records the plea "We only serve them in order that they may bring us nearer to Allah" (39:3) and answers it.
The Buddha told his hearers to be their own island and refuge (DN 16) and said one is the lord of oneself (Dhammapada 160). Prayer is therefore unnecessary: the work is ours.
Rebuttal
Muslims honour the call to effort. But no one makes himself or keeps himself alive, and the Qur'an says the One who does is close and answers every caller (2:186).
Pure Land and other Mahayana schools
Many Pure Land and other Mahayana Buddhists call on Amitabha or Avalokitesvara, on the strength of the Mahayana sutras. Some Pure Land teachers say saying the name is gratitude for a rescue already given, and some Mahayana readers take these figures as symbols. Many Zen practitioners pray to no one.
Mahayana, including Pure Land and Zen
Saying Amitabha's name is not a petition. It is gratitude for a rescue that his power (other-power) has already given. The vow promises rebirth in his land to those who turn to him.
Rebuttal
Gratitude to a rescuer is still addressed to one who hears. The vow promises this to those who turn to him in countless worlds. Only Allah (﷾) hears every caller and saves (40:60; 35:13-14).
The Lotus Sutra says that those who hear the name of Avalokitesvara, or implore him, are released from troubles. Calling on a bodhisattva of compassion is how compassion reaches us.
Rebuttal
The sutra promises rescue from fire and flood to anyone who calls him. The Qur'an teaches that those invoked besides Allah (﷾) cannot hear or answer, and that all dominion is His (35:13-14).
Amitabha and Avalokitesvara are symbols of the Buddha-nature in every mind, not persons who hear us. To call on them is to wake that nature.
Rebuttal
The sutras speak of a real land and a real rescue still to come. If the figures are symbols, nothing hears; if persons, they do what only Allah (﷾) does.
Many Zen practitioners pray to no one. Sitting in meditation and doing right are the whole of the practice, and the Pali Buddha already says the work is ours.
Rebuttal
Then this page does not say they pray, and Islam honours the effort. The Qur'an says hearts find rest in the remembrance of Allah (﷾), and it asks us to call on Him alone (13:28; 1:5).
The Islamic perspective
Both teachers point away from themselves. The Pali Buddha says the Buddhas only show the way. The Prophet (ﷺ) is told to say that he only invokes his Lord. Islam says to whom. Allah (﷾) hears and answers, no one is to be invoked along with Him, and intercession is only by His leave. Belief, worship and character together make one way of life.
The Qur'an has the Prophet (ﷺ) say that he does no more than invoke his Lord. It is not in his power, he says, to cause harm or to bring anyone to right conduct.
Allah (﷾) says, "Call on Me; I will answer your (Prayer)", and says that He is close and listens to every suppliant. Those invoked besides Him cannot hear or answer.
The Qur'an says that the places of worship are for Allah (﷾). So no one is to be invoked along with Him.
No one may intercede with Allah (﷾) except after His leave, and then only for those He accepts. This is Islam's own rule, stated without exception.
The Qur'an records the plea "We only serve them in order that they may bring us nearer to Allah". It answers that sincere devotion is due to Allah (﷾) alone.
The believer asks Allah (﷾) alone for help, and prayer, belief and character together make one way of life.