Short answer
Islam has no monks because God (﷾) never commanded monasticism, a life as a monk. The Qur'an says people invented it for themselves. Islam has scholars and prayer leaders, but no priests and no monks. None of them is asked to stay unmarried. Three men once vowed to pray all night, fast every day and never marry. The Prophet (ﷺ) told them that he prayed and slept, fasted and ate, and married. Whoever turns away from his way, he said, is not of him. He also refused a companion who asked to give up marriage. The Qur'an says God (﷾) gave the earlier messengers wives and children. In the Bible too, Peter, a disciple of Jesus, had a wife. Paul, a Christian missionary of the first century, says other church leaders took wives too. So Islam asks for a whole life lived in the world, with worship, work and family together.
Full explanation
What the Qur'an says about monasticism
The Qur'an says God (﷾) gave Jesus the Injil (the message God (﷾) revealed to Jesus) and put compassion and mercy in the hearts of his followers. Monasticism (rahbaniyya), a life lived apart from the world as a monk, was something they invented for themselves. God (﷾) did not prescribe it. He commanded only that they seek His good pleasure, and even that they did not keep as they should have done. The Qur'an also forbids believers to make unlawful the good things God (﷾) has made lawful. And it says God (﷾) gave the messengers before the Prophet (ﷺ) wives and children.
“Then, in their wake, We followed them up with (others of) Our messengers: We sent after them Jesus the son of Mary, and bestowed on him the Gospel; and We ordained in the hearts of those who followed him Compassion and Mercy. But the Monasticism which they invented for themselves, We did not prescribe for them: (We commanded) only the seeking for the Good Pleasure of Allah; but that they did not foster as they should have done. Yet We bestowed, on those among them who believed, their (due) reward, but many of them are rebellious transgressors.”
“O ye who believe! make not unlawful the good things which Allah hath made lawful for you, but commit no excess: for Allah loveth not those given to excess.”
“We did send messengers before thee, and appointed for them wives and children: and it was never the part of a messenger to bring a sign except as Allah permitted (or commanded). For each period is a Book (revealed).”
What the Prophet (ﷺ) taught about marriage
Three men once vowed to pray all night, fast without a break and never marry. The Prophet (ﷺ) told them he feared God (﷾) more than they did. Yet he prayed and slept, fasted and ate, and married. Whoever turns from his way, he said, is not of him. He refused Uthman ibn Maz'un, one of his companions, permission to give up marriage. And he told young men that whoever is able to marry should marry.
Anas ibn Malik reports that three men asked about the Prophet's (ﷺ) worship and thought their own too little. One vowed to pray all night for ever, one to fast every day without a break, and one to keep away from women and never marry. The Prophet told them that he feared Allah more than they did, yet he fasted and broke his fast, prayed and slept, and married. Whoever turns away from his way, he said, is not of him.
Sa'd ibn Abi Waqqas reports that the Prophet (ﷺ) refused Uthman ibn Maz'un permission to give up marriage, and that had he allowed it, they would have had themselves castrated.
Abdullah ibn Mas'ud reports that the Prophet (ﷺ) told the young men with him: whoever of you is able to marry should marry, because it helps him lower his gaze and guard his chastity; and whoever cannot should fast.
Marriage as a sign of God (﷾)
The Qur'an counts marriage among God's (﷾) signs. He made spouses from among ourselves, so that we may find rest in them, and put love and mercy between them. In Islam marriage is part of worship, not a step away from it.
“And among His Signs is this, that He created for you mates from among yourselves, that ye may dwell in tranquillity with them, and He has put love and mercy between your (hearts): verily in that are Signs for those who reflect.”
Why some churches ask for celibacy
Some Christians value celibacy, which means staying unmarried. They point to Jesus' words about those who make themselves eunuchs for the kingdom of heaven, which means those who give up marriage for God (﷾). They also point to Paul, who wished that all were unmarried as he was. The Catholic and Orthodox churches have monks and nuns who vow celibacy. In the Latin Catholic Church, the Catholic Church of the West, priests must as a rule stay celibate, by church law. Orthodox bishops are chosen only from unmarried men. Orthodox and Eastern Catholic priests may marry before they are ordained, which means made priests. Protestant ministers may marry.
“For there are some eunuchs, which were so born from their mother’s womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven’s sake. He that is able to receive it, let him receive it.”
“For I would that all men were even as I myself. But every man hath his proper gift of God, one after this manner, and another after that. I say therefore to the unmarried and widows, It is good for them if they abide even as I.”
The Catholic Church's law for the Latin Church binds its clergy, as a rule, to perfect and perpetual continence for the sake of the kingdom of heaven, and so to celibacy.
What the Bible says about Peter and other church leaders
Jesus said of celibacy, "He that is able to receive it, let him receive it". That makes it a gift for some, not a rule for anyone. Paul says the same, that every man has his own gift from God (﷾). Peter, whom Catholics hold to be the first Pope, had a wife. Jesus healed his wife's mother. Paul says that Peter and the other leaders of the early church took their wives with them. And the letter to Timothy lets a bishop be a married man, the husband of one wife. Jesus did not marry, but he never made celibacy a rule for his followers. His chief disciple was a married man. A rule that no priest may marry is not found in the Bible.
“For there are some eunuchs, which were so born from their mother’s womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven’s sake. He that is able to receive it, let him receive it.”
“For I would that all men were even as I myself. But every man hath his proper gift of God, one after this manner, and another after that. I say therefore to the unmarried and widows, It is good for them if they abide even as I.”
“And when Jesus was come into Peter’s house, he saw his wife’s mother laid, and sick of a fever.”
“Have we not power to lead about a sister, a wife, as well as other apostles, and as the brethren of the Lord, and Cephas?”
“A bishop then must be blameless, the husband of one wife, vigilant, sober, of good behaviour, given to hospitality, apt to teach;”
Verse by verse
Matthew 19:12 — Gospel of Matthew, chapter 19, verse 12
“For there are some eunuchs, which were so born from their mother’s womb: and there are some eunuchs, which were made eunuchs of men: and there be eunuchs, which have made themselves eunuchs for the kingdom of heaven’s sake. He that is able to receive it, let him receive it.”
Cited for
Jesus honours those who give up marriage for the kingdom of heaven, so a life of celibacy is a higher way.
The response
Jesus ends the saying with "He that is able to receive it, let him receive it". He makes it a matter for the few who can bear it, not a law. His chief disciple Peter was married.
Where it lands — The verse allows celibacy for some. It does not require it of any church leader.
1 Corinthians 7:7-8 — First Epistle to the Corinthians, chapter 7, verses 7-8
“For I would that all men were even as I myself. But every man hath his proper gift of God, one after this manner, and another after that. I say therefore to the unmarried and widows, It is good for them if they abide even as I.”
Cited for
Paul stayed unmarried and wished others would do the same.
The response
Paul himself, a Christian missionary of the first century, says in the same passage that every man has his own gift from God (﷾). Elsewhere he says that Peter and the other leaders of the early church took wives with them. The letter to Timothy lets a bishop be a married man and asks only that he be the husband of one wife.
Where it lands — The New Testament treats marriage as normal for church leaders and bishops, and it never forbids it to them.
Perspectives
Catholic and Orthodox
Celibacy freely chosen for the kingdom of heaven is a high calling. Monks and nuns vow it. The Latin Catholic Church asks it of its priests.
Jesus praised those who make themselves eunuchs for the kingdom of heaven's sake.
Rebuttal
Jesus does speak of those who give up marriage for the kingdom of heaven. He calls it a gift for those able to receive it. A gift for some cannot become a rule for a whole priesthood, and his chief disciple Peter was married. Islam goes further: the Prophet (ﷺ) refused even a companion who asked permission to give up marriage.
Paul wished that all were unmarried as he was.
Rebuttal
Paul did say he wished that all were as he was. But in the same passage he says every man has his own gift from God (﷾). He also says that Peter and the other leaders of the early church took wives with them. The letter to Timothy lets a bishop be a husband. The Prophet (ﷺ), who told three men seeking extra worship that he feared God (﷾) more than they did, married. In Islam marriage is part of worship, not a rival to it.
The Latin Catholic Church binds its clergy, as a rule, to celibacy.
Rebuttal
Catholics agree this is a church rule, not a command of the Bible. But the Qur'an forbids making unlawful the good things God (﷾) has made lawful. It says monasticism was invented by people and not prescribed by God (﷾). Marriage is something God (﷾) made lawful, so no church may forbid it to those who serve Him.
The Islamic perspective
God (﷾) never prescribed monasticism. The Prophet (ﷺ) married and forbade his followers to give up marriage for worship. He said that whoever turns from his way is not of him.
The Qur'an says monasticism was invented by people and not prescribed by God (﷾).
The Qur'an forbids believers to make unlawful what God (﷾) has made lawful.
The Prophet (ﷺ) rejected a vow never to marry and said whoever turns from his way is not of him.
The Qur'an says God (﷾) sent messengers before the Prophet (ﷺ) and gave them wives and children.
Peter had a wife. Paul says that Peter and the other leaders of the early church took wives with them. A bishop was to be the husband of one wife.