Short answer
No. Jesus said he came not to destroy the Law of Moses but to fulfil it. He said not one jot (the smallest letter in Hebrew writing) would pass until heaven and earth pass. He warned that whoever breaks the least commandment, and teaches others to, will be called least in the kingdom of heaven. He was circumcised, and it was his custom to worship in the synagogue on the Sabbath. He told a man who asked about eternal life to keep the commandments. Years later, his disciples were still arguing over whether converts must keep the Law. It was Paul, a Christian missionary of the first century, who told Christians they are "not under the law". Islam's answer: the Qur'an says Jesus came to confirm the Law, and to make lawful some of what had been forbidden. Only God (﷾) changes His Law, through a prophet.
Full explanation
The verses Christians rely on
The Law of Moses is the set of commandments God (﷾) gave to Moses. Christians read "fulfil" in Matthew 5:17 to mean that Jesus completed what the Law pointed to, so that its rules no longer bind. They support this from Paul, who calls Christ the end of the law. They also point to Mark 7:19, where the words "making all foods clean" appear. Many modern translations print these words as Mark's note that Jesus declared all foods clean.
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.”
“For Christ is the end of the law for righteousness to every one that believeth.”
“because it doesn’t go into his heart, but into his stomach, then into the latrine, making all foods clean?””
What Jesus said about the Law
Jesus did not stop at "fulfil". In the next verse he says that not one jot will pass from the Law until heaven and earth pass. In the verse after, he says that whoever breaks the least commandment and teaches men to do so will be least in the kingdom. Luke gives the same saying. When a man asked how to enter into life, Jesus told him to keep the commandments.
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.”
“For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.”
“Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.”
“And it is easier for heaven and earth to pass, than one tittle of the law to fail.”
“And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.”
Jesus kept the Law himself
Jesus was circumcised on the eighth day, as the Law commands. Circumcision is the cutting of the foreskin, the sign of God's (﷾) covenant with Abraham. It was Jesus' custom to go to the synagogue, the Jewish place of worship, on the Sabbath. The Sabbath is the weekly day of rest. The Law calls circumcision an everlasting covenant and the Sabbath a perpetual covenant. It calls the ban on eating fat and blood a perpetual statute. These were rules for Israel. In Islam, only God (﷾) lifts His rules, and only through a prophet, as He later did through the Prophet Muhammad (ﷺ).
“And when eight days were accomplished for the circumcising of the child, his name was called JESUS, which was so named of the angel before he was conceived in the womb.”
“And he came to Nazareth, where he had been brought up: and, as his custom was, he went into the synagogue on the sabbath day, and stood up for to read.”
“He that is born in thy house, and he that is bought with thy money, must needs be circumcised: and my covenant shall be in your flesh for an everlasting covenant.”
“Wherefore the children of Israel shall keep the sabbath, to observe the sabbath throughout their generations, for a perpetual covenant.”
“It shall be a perpetual statute for your generations throughout all your dwellings, that ye eat neither fat nor blood.”
“For those who followed the Jewish Law, We forbade every (animal) with undivided hoof, and We forbade them that fat of the ox and the sheep, except what adheres to their backs or their entrails, or is mixed up with a bone: this in recompense for their wilful disobedience: for We are true (in Our ordinances).”
“"Those who follow the messenger, the unlettered Prophet, whom they find mentioned in their own (scriptures),- in the law and the Gospel;- for he commands them what is just and forbids them what is evil; he allows them as lawful what is good (and pure) and prohibits them from what is bad (and impure); He releases them from their heavy burdens and from the yokes that are upon them. So it is those who believe in him, honour him, help him, and follow the light which is sent down with him,- it is they who will prosper."”
The first disciples still kept the Law
Years after Jesus had gone, believers from Judea were still teaching that Gentile converts must be circumcised and keep the Law of Moses. Gentiles are people who are not Jewish. The apostles met to argue the question. That debate would hardly have been needed if Jesus had settled it. They ruled that Gentile converts need not be circumcised. But they quoted no saying of Jesus for it, and the Jewish believers went on keeping the Law. Peter, who had followed Jesus, said he had never eaten anything unclean. The Jerusalem elders told Paul that thousands of Jewish believers were all zealous for the Law. They said they had heard he taught Jews to forsake Moses.
“And certain men which came down from Judea taught the brethren, and said, Except ye be circumcised after the manner of Moses, ye cannot be saved.”
“But there rose up certain of the sect of the Pharisees which believed, saying, That it was needful to circumcise them, and to command them to keep the law of Moses.”
“Wherefore my sentence is, that we trouble not them, which from among the Gentiles are turned to God:”
“But Peter said, Not so, Lord; for I have never eaten any thing that is common or unclean.”
“And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law:”
“And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs.”
The break with the Law came after Jesus, above all from Paul
It is Paul, not Jesus, who tells Christians they are not under the law. It is Paul who says the law of commandments was abolished. It is Paul who warns the believers of Galatia (a region of Asia Minor) that if they are circumcised, Christ will profit them nothing. He tells believers to let no one judge them over food, drink or the Sabbath. He says every creature of God (﷾) is good and nothing is to be refused, while the Law calls some foods unclean. Yet the same Paul circumcised Timothy, a young disciple who worked with him. Paul also says a man is justified (made right with God (﷾)) by faith without the deeds of the law. The letter of James says a man is justified by works and not by faith only.
“For sin shall not have dominion over you: for ye are not under the law, but under grace.”
“Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of twain one new man, so making peace;”
“Behold, I Paul say unto you, that if ye be circumcised, Christ shall profit you nothing.”
“Him would Paul have to go forth with him; and took and circumcised him because of the Jews which were in those quarters: for they knew all that his father was a Greek.”
“Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:”
“For every creature of God is good, and nothing to be refused, if it be received with thanksgiving:”
“And the swine, though he divide the hoof, and be clovenfooted, yet he cheweth not the cud; he is unclean to you.”
“Therefore we conclude that a man is justified by faith without the deeds of the law.”
“Ye see then how that by works a man is justified, and not by faith only.”
What the Qur'an says about Jesus and the Law
The Qur'an says Jesus came to confirm the Law that was before him, and to make lawful some of what had been forbidden to Israel. It does not say he cancelled it. It says some good things had been forbidden to Israel for their wrongdoing. Lifting part of that burden was a relief given by the One who gave the Law, inside the Law. It did not declare the Law over. His message was to the Children of Israel, and it stood inside the Law of Moses. God (﷾) gave each people a law and a way, and through the Prophet Muhammad (ﷺ) He lifted Israel's heavy burdens.
“"'(I have come to you), to attest the Law which was before me. And to make lawful to you part of what was (Before) forbidden to you; I have come to you with a Sign from your Lord. So fear Allah, and obey me.”
“For the iniquity of the Jews We made unlawful for them certain (foods) good and wholesome which had been lawful for them;- in that they hindered many from Allah's Way;-”
“And in their footsteps We sent Jesus the son of Mary, confirming the Law that had come before him: We sent him the Gospel: therein was guidance and light, and confirmation of the Law that had come before him: a guidance and an admonition to those who fear Allah.”
“And remember, Jesus, the son of Mary, said: "O Children of Israel! I am the messenger of Allah (sent) to you, confirming the Law (which came) before me, and giving Glad Tidings of a Messenger to come after me, whose name shall be Ahmad." But when he came to them with Clear Signs, they said, "this is evident sorcery!"”
“To thee We sent the Scripture in truth, confirming the scripture that came before it, and guarding it in safety: so judge between them by what Allah hath revealed, and follow not their vain desires, diverging from the Truth that hath come to thee. To each among you have we prescribed a law and an open way. If Allah had so willed, He would have made you a single people, but (His plan is) to test you in what He hath given you: so strive as in a race in all virtues. The goal of you all is to Allah; it is He that will show you the truth of the matters in which ye dispute;”
“"Those who follow the messenger, the unlettered Prophet, whom they find mentioned in their own (scriptures),- in the law and the Gospel;- for he commands them what is just and forbids them what is evil; he allows them as lawful what is good (and pure) and prohibits them from what is bad (and impure); He releases them from their heavy burdens and from the yokes that are upon them. So it is those who believe in him, honour him, help him, and follow the light which is sent down with him,- it is they who will prosper."”
What Islam kept
Islam still forbids blood and the flesh of swine, as the Law of Moses did. The Prophet (ﷺ) counted circumcision first among the practices of the natural way. The everlasting covenant that Jesus kept in his own body is still kept by Muslims today.
“He hath only forbidden you dead meat, and blood, and the flesh of swine, and that on which any other name hath been invoked besides that of Allah. But if one is forced by necessity, without wilful disobedience, nor transgressing due limits,- then is he guiltless. For Allah is Oft-forgiving Most Merciful.”
“And the swine, though he divide the hoof, and be clovenfooted, yet he cheweth not the cud; he is unclean to you.”
Abu Hurayra reports that the Prophet (ﷺ) named circumcision first among five practices of the natural way (fitra).
Verse by verse
Matthew 5:17 — Gospel of Matthew, chapter 5, verse 17
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.”
Cited for
Jesus came to fulfil the Law, and once it was fulfilled it no longer binds his followers.
The response
The very next verse sets the limit: not one jot passes from the Law "till heaven and earth pass". The verse after it calls anyone who breaks the least commandment, and teaches others to, least in the kingdom. So Jesus explained his own word, and he did not explain it as an ending. Christians point to the last words, "till all be fulfilled", and say all was fulfilled at the cross. But Jesus set the first limit by heaven and earth. In the next verse he tells his hearers to keep and teach even the least commandments. Luke gives the same saying with no "fulfilled" clause at all. It says it is easier for heaven and earth to pass than one tittle (the smallest stroke of a letter) of the law to fail. Luke places it straight after Jesus says "the law and the prophets were until John". So even that saying does not mean the Law has ended.
Where it lands — Jesus never said the Law was over. Only God (﷾) changes His Law, through a prophet, and Paul brought no revelation from Him.
John 5:18 — Gospel of John, chapter 5, verse 18
“Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.”
Cited for
Jesus broke the Sabbath, so he did not bind himself to the Law.
The response
John's words give the Sabbath charge on which they sought to kill him: he had healed a man on the Sabbath. When the same charge was made in Matthew, Jesus answered that "it is lawful to do well on the sabbath days". And it was his custom to worship in the synagogue on the Sabbath. He rejected the added rules of the scribes (the Jewish teachers of the Law), not the commandment.
Where it lands — Jesus disputed how the Sabbath was kept, never whether it was kept.
Romans 10:4 — Epistle to the Romans, chapter 10, verse 4
“For Christ is the end of the law for righteousness to every one that believeth.”
Cited for
Paul teaches that Christ is the end of the law, so Christians are free of it.
The response
These are Paul's words, not Jesus' words. Paul did not follow Jesus during his ministry. The believers in Jerusalem who had known Jesus were still zealous for the Law. They were troubled by reports that Paul taught Jews to forsake Moses.
Where it lands — The end of the Law is Paul's teaching. Jesus never taught it.
Mark 7:19 — Gospel of Mark, chapter 7, verse 19
“because it doesn’t go into his heart, but into his stomach, then into the latrine, making all foods clean?””
Cited for
In Mark 7:19 Jesus declares all foods clean, so the food laws are over.
The response
The King James Version and the World English Bible both print these words as part of Jesus' description of digestion. Some modern translations turn them into Mark's comment, "he declared all foods clean". The dispute in the chapter is about eating with unwashed hands, and Matthew's telling of the same scene closes on exactly that point. So when Jesus says "there is nothing from without a man, that entering into him can defile him", he is answering the washing question. And Peter, who asked Jesus about this saying, still said years later that he had never eaten anything unclean. Even the vision he later saw is explained in Acts as being about people, not food. God (﷾) had shown him that he should not call any man common or unclean.
Where it lands — Peter heard Jesus speak, and years later he still kept the food laws. Jesus had not ended them.
Galatians 5:2 — Epistle to the Galatians, chapter 5, verse 2
“Behold, I Paul say unto you, that if ye be circumcised, Christ shall profit you nothing.”
Cited for
Paul says circumcision is of no use to those in Christ.
The response
Genesis calls circumcision an everlasting covenant, and Jesus was circumcised. Yet Paul himself took Timothy and circumcised him.
Where it lands — The covenant Jesus kept in his own body was set aside by Paul for Gentile believers. Paul still circumcised Timothy when it helped him among the Jews.
Perspectives
Mainstream Christian
Jesus fulfilled the Law in his life and death. Believers are now under grace, not under the Law of Moses, though its moral teaching still guides them.
Paul writes that Christians are not under the law but under grace.
Rebuttal
Paul does write this. But these are Paul's words, not Jesus' words, and Jesus said not one jot would pass until heaven and earth pass. The believers in Jerusalem who had known Jesus were still zealous for the Law.
Paul calls Christ the end of the law for righteousness to everyone who believes.
Rebuttal
Paul does write this, but Jesus never said it. Jesus told his hearers to keep and teach even the least commandment. So "the end of the law" is Paul's teaching, not the teaching of Jesus.
Christians read "fulfil" in Matthew 5:17 as completing the Law, so that its ceremonies no longer bind.
Rebuttal
The next verse sets the limit: not one jot passes until heaven and earth pass. Christians say "till all be fulfilled" came at the cross, but Jesus set the first limit by heaven and earth. Luke gives the same saying with no "fulfilled" clause.
The Islamic perspective
Jesus confirmed the Law of Moses and lived by it. He made lawful some of what had been forbidden, but he never abolished it. The Law was set aside later, by Paul.
Jesus said that not one jot or tittle would pass from the Law until heaven and earth pass.
Jesus spoke of whoever breaks the least commandment and teaches others to do so. Such a person, he said, will be called least in the kingdom of heaven.
Jesus told a man seeking eternal life to keep the commandments.
Jesus was circumcised and worshipped on the Sabbath, both of which the Law calls lasting covenants.
Years after Jesus, the apostles were still debating whether converts must keep the Law. Jewish believers remained zealous for it.
No saying of Jesus frees his followers from the Law. The statements that Christians are not under the law come from Paul, and the Jerusalem ruling came years after Jesus.
The Qur'an teaches that Jesus came to confirm the Law and to make lawful part of what had been forbidden.