Short answer
There can only be one God, because two gods could not both be God. Two all-powerful beings would limit each other, and a god who can be limited is not God. Suppose their wills could ever clash. Then one would have to give way, and the one that gave way was never God. Suppose instead they could never disagree about anything. Then each is bound by the other, so neither is all-powerful. They are really one being with two names. The Qur'an makes this argument itself. It says that if there were other gods besides Allah (﷾), the heavens and the earth would fall into disorder. The universe shows the opposite. The same laws hold in the farthest galaxies our telescopes can see as in this room. One order points to one will. Islam's answer is simple: God (﷾) is One, and nothing is like Him.
Full explanation
Two all-powerful beings cannot exist
Follow the reasoning step by step. Two beings cannot both be all-powerful. Each would be limited by what the other could do, and a god who can be limited is not God. Suppose two divine wills could ever disagree. Then one would have to give way, and the one that gave way was never God. Suppose they could never disagree about anything at all. Then neither is free to will otherwise. Each is bound by the other, so neither is all-powerful. Two beings that can never differ are in the end one being with two names. Every road ends at one.
“No son did Allah beget, nor is there any god along with Him: (if there were many gods), behold, each god would have taken away what he had created, and some would have lorded it over others! Glory to Allah! (He is free) from the (sort of) things they attribute to Him!”
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
Suppose there were two gods.
Could both be all-powerful?
No. Each would be limited by what the other could do, and a god who can be limited is not God.
Could their wills ever clash?
If they could, one would have to give way, and the one that gave way was never God.
What if they could never disagree?
Then neither is free to will otherwise. They are not two gods, but one being with two names.
What does the universe show?
One order: the same laws hold as far as our telescopes can see. The Qur'an says that if there were other gods besides Allah (﷾), the heavens and the earth would fall into disorder.
One order points to one will, and one God (﷾).
Two beings that need nothing cannot exist
The argument from contingency reaches the same place by another road. It starts from the fact that everything we see depends on something else. It ends at a being that depends on nothing and has no parts. Muslim philosophers called this being wajib al-wujud (the being whose existence is necessary). Now suppose there were two such beings. They would have to differ in something, or they would not be two. Whatever made them differ would be a part or a limit. A being that depends on nothing can have neither. Ibn Sina, a Muslim philosopher, set out this reasoning in the eleventh century.
Divides all that exists into what is only possible in itself (mumkin al-wujud), which needs a cause, and what is necessary in itself (wajib al-wujud), which needs none. Argues that a chain of possible things, however long, needs a necessary being outside it, and that this being is one and has no parts.
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
The Qur'an gives reasons, not only a claim
The Qur'an does not simply say that God (﷾) is one. It gives reasons. If there were other gods, heaven and earth would fall into disorder. Each god would take away what he had made, and some would try to rule over the others. Any other god would have sought out a way to the Lord of the Throne, who alone holds it. Divided power means rivalry, and rivalry means ruin.
“If there were, in the heavens and the earth, other gods besides Allah, there would have been confusion in both! but glory to Allah, the Lord of the Throne: (High is He) above what they attribute to Him!”
“No son did Allah beget, nor is there any god along with Him: (if there were many gods), behold, each god would have taken away what he had created, and some would have lorded it over others! Glory to Allah! (He is free) from the (sort of) things they attribute to Him!”
“Say: If there had been (other) gods with Him, as they say,- behold, they would certainly have sought out a way to the Lord of the Throne!”
Many gods, many quarrels
The old religions of many gods often tell the very story the Qur'an describes. In the Greek myths the gods quarrel, take sides, deceive one another and are overthrown. Zeus seizes power from his own father, Kronos. Wherever there is more than one god, authority is divided. Our universe shows no such split. The same few laws hold at the farthest reach of our instruments as hold on earth. They have held as far back in time as we can observe. The Qur'an invites the reader to look for a flaw in creation and find none.
“He Who created the seven heavens one above another: No want of proportion wilt thou see in the Creation of (Allah) Most Gracious. So turn thy vision again: seest thou any flaw?”
“No son did Allah beget, nor is there any god along with Him: (if there were many gods), behold, each god would have taken away what he had created, and some would have lorded it over others! Glory to Allah! (He is free) from the (sort of) things they attribute to Him!”
Two wills in Jesus' prayer
Now apply the same test to the New Testament Gospels. In Luke, Jesus prays on the Mount of Olives that the cup (the suffering he dreaded) be taken from him. Then he yields to the Father's will. Matthew and Mark name the place Gethsemane. There are two wills in that prayer. They do not want the same thing, and one gives way. By the reasoning above, the one that gave way was never God. Jesus himself called "the Lord our God is one Lord" the first of all the commandments.
“Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.”
“And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:”
The Christian reply
Luke 22:42 — Gospel of Luke, chapter 22, verse 42
“Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.”
Cited for
The one person of Christ has two wills, divine and human. In Gethsemane he submitted according to his human will.
The response
Christians say a will belongs to a nature, not to a person. The Third Council of Constantinople, a meeting of bishops in 681, made this the official teaching. But the words "not my will" still have a speaker. The Council of Chalcedon (451) says that speaker is one person, God the Son. So the one person the creeds call God asked for something the Father did not grant, and yielded: "not my will, but thine". Christians also say his divine will is the Father's will. Then one person both willed the cup and asked to be spared it. Two natures move the problem to the one person; they do not remove it. One who must give way to God (﷾) is not God.
Where it lands — The prayer shows a servant submitting to his Lord, which is what the Qur'an says Jesus was.
Perspectives
Mainstream Christian
Christians also confess one God. They hold that the one God is three persons sharing one divine will. In the prayer at Gethsemane, they say, it was the human will of Christ that submitted to the Father.
Jesus taught that the first commandment is that the Lord our God is one Lord.
Rebuttal
Granted: Christians and Muslims agree that God (﷾) is one. The question is what "one" means. Jesus gave the commandment as "one Lord" and never added that the one God is three persons. In John 17:3 he calls the Father "the only true God".
Christians read Luke 22:42 as the human will of Christ submitting to the divine will. They say the one person of Christ has two natures and two wills.
Rebuttal
Granted: Christians say a will belongs to a nature, and his divine will is the Father's will. But "not my will" has one speaker, whom the creeds call God the Son. So one person both chose the cup and asked to be spared it. And one who must give way to God (﷾) is not God.
Buddhist: many gods, none of them eternal
Buddhism argues neither for one God nor for many. Its texts speak of many gods, all born and all passing, and none of them the maker of the world.
The gods are beings inside the cycle of rebirth. A god who thinks he is eternal is wrong, because he too will pass away.
Rebuttal
Islam agrees that no being who is born and passes away is God. The question left is what holds up the whole round of passing things. A being that depends on nothing has no parts and no limit, and two such beings could not differ in anything, so there can only be one. The Qur'an says: He is Allah (﷾), the One and Only.
Chinese tradition: a god for each need
Chinese popular religion honours many gods. There is the Kitchen God at the family stove, the god of wealth, Mazu the goddess of sailors, and the family's own ancestors. People turn to whichever one fits their need.
Different gods look after different parts of life, so it is natural to honour many.
Rebuttal
Gods who each look after one part of life are each limited, and a limited god is not God. The Qur'an says that if there were many gods, each would take away what he had created, and some would rule over others. One unbroken order in the world points to one Lord over all of it.
Confucian: Heaven, ancestors and spirits together
The Confucian classics speak of Heaven, and of Shangdi, the Supreme Emperor. They also speak of many spirits who were given sacrifices. Confucians honour Heaven, the ancestors and the spirits together.
The wise king Shun made sacrifices to Shangdi, and also to the hills, the rivers and all the spirits. So honouring many besides Heaven is the old way.
Rebuttal
The same text names Shangdi first. And Fung Yu-lan notes that the spirits ranked below the Supreme Emperor. Islam gives worship to the One alone. The places of worship are for Allah (﷾), so call on no one beside Him.
Daoist: the many come from the one
Daoist temples honour many gods, such as the Three Pure Ones and the Jade Emperor. Yet the Tao Te Ching teaches that all things come from the one Tao.
Many deities may be honoured, because the many all come from the one Tao.
Rebuttal
Then only the one source is first, and everything it produced depends on it. Worship belongs to the One who gives, not to what was given. The Qur'an says: take not two gods for worship, for He is just One Allah (﷾).
One God was a later idea
Many atheists and scholars of religion say that belief in one God grew out of belief in many gods. On this view, believing in one God is a late stage in human religion, not a truth that reason requires.
Polytheism was the first religion of mankind; belief in one God came later.
Rebuttal
Hume, who argued this, drew the line himself. He traced the many gods to people's hopes and fears about the events of life, not to the study of nature. And he wrote that anyone led to God (﷾) by the works of nature could only think of one single being, because the parts of the universe fit one plan. So even on his account, reason points to one.
Shinto: the kami are many
Shinto speaks of countless kami, or spirits. The first three kami in the Kojiki came into being together. Shinto scholars say kami is a different idea from the one God of Judaism and Christianity.
Kami are many by nature. Anything that fills us with awe may be kami, so there is no reason to look for only one.
Rebuttal
Then the kami are many because they are things that came into being. And the awe they inspire should reach the One who made them. The Kojiki says the first kami were born when Heaven and Earth began. Whatever comes into being is not the One who has no beginning. The Qur'an says that if there were other gods besides Allah (﷾), there would be disorder in heaven and on earth.
The Islamic perspective
God (﷾) is one, and He can only be one. Reason shows it, the order of the universe shows it, and the Qur'an argues it in plain words.
The Qur'an argues from the idea of other gods. If there were any, heaven and earth would fall into disorder, and the gods would fight one another.
Two all-powerful beings would limit each other, and a god who can be limited is not God.
If two divine wills could clash, the one that gave way was never God. If they could never clash, each is tied to the other, and they are one being with two names.
Two beings that each depended on nothing would have to differ in some part or limit. But a being like that cannot have parts or limits.
One unbroken order of nature points to one will behind it.
Allah (﷾) is One, begets not and is not begotten, and there is none like Him.
Luke records two wills in Gethsemane that do not want the same thing, and one of them gives way.