Short answer
No. The Qur'an says his enemies did not kill Jesus or crucify him, so no curse ever touched him. In the King James Bible, the Law of Moses calls a man hanged on a tree accursed of God (﷾). Paul, a Christian missionary of the first century, quotes that verse and accepts it. He writes that Christ was made a curse for us. So the Christian story needs God's (﷾) own Law to curse His Messiah. The Qur'an gives Jesus the opposite place. It says he is held in honour in this world and the next, and is one of those nearest to God (﷾). God (﷾) raised him up to Himself. Jews objected to a crucified Messiah from the beginning, and Paul called the cross a stumbling block to them. Islam's answer is that the curse belongs to the Christian story of the cross. The Jesus of the Qur'an was never under it.
Full explanation
What the Law of Moses says
The Law of Moses is the first five books of the Bible. Its book of Deuteronomy speaks of a man put to death for a sin worthy of death and then hanged on a tree. His body must be buried the same day, because, in the King James wording, the hanged man is accursed of God (﷾).
“And if a man have committed a sin worthy of death, and he be to be put to death, and thou hang him on a tree:”
“His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the Lord thy God giveth thee for an inheritance.”
What Christians believe
Christians hold that Jesus, though innocent, took the curse of the Law on himself. They say he did this so that others could be freed from it. Paul says this in Galatians, quoting the Law's own words. In Acts, Peter and the other first disciples speak of Jesus as the one who was slain and hanged on a tree.
“Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree:”
“The God of our fathers raised up Jesus, whom ye slew and hanged on a tree.”
“And we are witnesses of all things which he did both in the land of the Jews, and in Jerusalem; whom they slew and hanged on a tree:”
The objection is as old as the church
Paul admits that a crucified Messiah, the promised king whom God (﷾) would send, was a stumbling block to the Jews. In the second century Justin Martyr, a Christian writer, wrote a dialogue with a Jew named Trypho. Trypho's objection rests on this law. The law calls the crucified man accursed, so how could the Messiah die that way?
“But we preach Christ crucified, unto the Jews a stumblingblock, and unto the Greeks foolishness;”
In this second-century Christian work, the Jewish speaker Trypho says that his people doubt the Messiah could be so shamefully crucified, because the Law calls whoever is crucified accursed.
Muslim scholars made the same argument
Christine Schirrmacher, a Christian scholar of Islam, reports that Muslim theologians argue from the Bible itself here. The Law says a man hung on a tree is under God's (﷾) curse. Jesus was an honoured prophet, not a cursed criminal. So, they say, he could not have died on a cross. She also quotes Hermann Stieglecker, another Christian scholar of Islam, who sums up the Muslim view. To say that God (﷾) let Himself be mocked and mistreated by the lowest mob, he writes, is an outrageous disgrace. So is saying that God (﷾) died a criminal's death between two criminals.
A Christian scholar of Islam reports that Muslim theologians argue from Deuteronomy 21:23 that Jesus, an honoured prophet and not a cursed criminal, could not have died on a cross, and that a shameful death would have been unworthy of a prophet.
A Christian scholar of Islam sums up the Muslim view: to say that God let Himself be so humbled that the lowest mob could mock and mistreat Him, and that He died the most shameful and painful death, as a criminal between two real criminals, is an outrageous disgrace.
“His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the Lord thy God giveth thee for an inheritance.”
What the Qur'an says of Jesus
The Qur'an states that Jesus is held in honour in this world and the Hereafter, and is among those nearest to Allah (﷾). It also states that his enemies did not kill him and did not crucify him. It was made to appear so to them, and Allah (﷾) raised him up to Himself. God (﷾) did not let His honoured messenger fall under the curse of His own Law. Matthew's account has the man on the cross cry, "My God, my God, why hast thou forsaken me?" So the Christian account has God (﷾) forsake His Messiah. The Qur'an has God (﷾) raise him up to Himself.
“Behold! the angels said: "O Mary! Allah giveth thee glad tidings of a Word from Him: his name will be Christ Jesus, the son of Mary, held in honour in this world and the Hereafter and of (the company of) those nearest to Allah;”
“That they said (in boast), "We killed Christ Jesus the son of Mary, the Messenger of Allah";- but they killed him not, nor crucified him, but so it was made to appear to them, and those who differ therein are full of doubts, with no (certain) knowledge, but only conjecture to follow, for of a surety they killed him not:-”
“Nay, Allah raised him up unto Himself; and Allah is Exalted in Power, Wise;-”
“And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?”
Verse by verse
Galatians 3:13 — Epistle to the Galatians, chapter 3, verse 13
“Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree:”
Cited for
Christians read this as Christ, who was innocent, willingly taking on the curse that belonged to others. They say this set the others free from it.
The response
Paul puts the curse meant for one put to death for his own sin on a man Christians call innocent. Through Ezekiel, God (﷾) says the son shall not bear the father's iniquity. The soul that sins is the one that dies. The Qur'an says no bearer of burdens bears another's burden. It also says God (﷾) honoured Jesus in this world and the Hereafter.
Where it lands — In God's (﷾) own judgement each soul bears its own sin, and the Qur'an's Jesus was never under the curse.
Perspectives
Mainstream Christian
Jesus was innocent, but on the cross he willingly became a curse in the place of sinners. Christians say this freed them from the curse of the Law.
Paul writes that Christ redeemed us from the curse of the Law by being made a curse for us. He quotes the Law on the man hanged on a tree.
Rebuttal
Paul does write this, and he quotes the Law. So he puts the curse meant for a sinner on a man Christians call innocent. Ezekiel says the son does not bear the father's sin, and the soul that sins is the one that dies.
In Acts, the first disciples speak of Jesus as one who was slain and hanged on a tree.
Rebuttal
Acts does say Jesus was "hanged on a tree", the very picture of the Law's curse in Deuteronomy. That ties him to the curse instead of lifting it. The Qur'an says his enemies did not kill him or crucify him, so the curse never touched him.
The Islamic perspective
Jesus is honoured by God (﷾) in this world and the next. God (﷾) did not let His messenger die the death His own Law calls accursed. His enemies did not kill him or crucify him, and God (﷾) raised him up to Himself.
The Law of Moses says that a man hanged on a tree is accursed of God (﷾).
Paul does not deny that the verse applies to Jesus. He says Christ was made a curse.
The Qur'an calls Jesus held in honour in this world and the Hereafter. It calls him one of those nearest to Allah (﷾).
The Qur'an states that they did not kill him and did not crucify him. Allah (﷾) raised him up to Himself.
An accursed death does not fit an honoured prophet. So the Christian account sets God's (﷾) Law against God's (﷾) Messiah.
Jews objected early that a crucified man falls under the Law's curse. Paul calls the cross a stumbling block to them. Justin's Trypho makes the objection in so many words.
Muslim theologians argue from Deuteronomy 21:23 that Jesus, an honoured prophet, could not have died on a cross.
Christian scholars of Islam record the Muslim view of such a death. A shameful death between two criminals is a disgrace God (﷾) would never suffer.