Here's Dr. Shabir Ally's answer. Watch the video response here
There’s no clear statement in the Quran to the effect that Jesus was sinless. However, there’s an interesting verse that says that when the angel came to Mary to deliver her the good news that she will be having a son, the angel said, “...So that I may confer on you a pure boy”. This statement could point to Jesus’ sinlessness, yet this is not conclusive since it can’t determine whether or not he remained pure throughout his life.
Being born pure was not exclusive to Jesus. All children are sinless by definition - they’re born pure, on what is known as a fitrah. Fitrah can be described as the child’s natural disposition to worship God. As children grow up and reach the age of discretion, they’re able to choose their paths and either continue worshipping God or turn away. The fact that this is mentioned about Jesus, in particular, is distinctive and we should pay attention to this. Sometimes in the Quran there is a mention of a prophet doing something which is not quite Islamic, but it’s excusable because of the circumstances or accredited to human nature -- this doesn't include gross sins like adultery. In the Bible, there are reports about prophets committing gross sins, like David committing adultery with Bethsheba. In the Quran, however, although his story is told at length, this detail is not there. Muslims are able to interpret the Quranic stories in a way that upholds the integrity and moral disposition of the prophets. However, as human beings, the prophets are prone to error.
In the case of Jesus, no such thing is mentioned. Jesus is depicted as very pure in the Quran, to the extent that on the Day of Judgement, Jesus will be called to witness against his people. So although Jesus’ people and followers erred and deviated from the teachings, Jesus himself is depicted as pure regardless.
Some may wonder why Jesus is depicted as pure throughout his life in the Quran when other prophets’ faults and blemishes are often shown. It could be that the Quran is reaching out with an olive branch to Christians. Since Jesus was held in such high esteem among our Christian communities, the Quran might have wanted to present Jesus in a way that Christians could relate and understand. The main difference is that Jesus is not God according to the Quran, but he is presented in a way that confers respect.
In the hadith literature, however, more indications help us to think of Jesus as being sinless. In one hadith, it is said that when a child is born, Satan touches that child and causes the child to cry. That’s the case of every child except for Jesus and his mother. In a way, this speaks to the immaculate conception of Mary, but it also suggests that Jesus was born in some way so that he was not touched by Satan.
More to the point is that on the Day of Judgement, it is said in Islamic tradition that people will be in turmoil, waiting for the judgement to begin, and then very anxious to know what their outcome will be. They will plead with their prophets to be able to intercede with God, asking Him to get the judgement over with. When they approach the prophets one after another, the prophets will confess to some mishap in their own lives, something that is considered to be less than ideal. The prophets will announce that they don’t feel comfortable approaching God after the confessions of their wrongdoings, and the people will promptly move on to the next prophet. They will keep going to one prophet after another until they come to Jesus. When they come to Jesus, Jesus will defer them to the Prophet Muhammad, but he will not mention anything that could be characterized as a sin.
In conclusion, we believe that prophets, in general, do not succumb to great moral failures. Prophets are human beings and they may make momentary slips which are forgivable and excusable. However, in the case of Jesus, there is no mention of such a slip.