Short answer
No. God (﷾) cannot die. The Bible calls God "immortal" (1 Timothy 1:17), the One "who only hath immortality" (1 Timothy 6:16). The Qur'an calls Him the One "Who lives and dies not" (25:58). Yet Christians believe that Jesus is God the Son, and that he died on the cross. Most churches say he died in his human nature, while his divine nature did not die. But a nature does not die; a person does. So either God (﷾) died, which the Bible denies, or only a man died. Islam's answer is simpler. God (﷾) is ever-living. Jesus was His messenger, and the Qur'an says he was not killed (4:157).
Full explanation
What Christians believe
Christians believe that Jesus is God the Son, and that he died on the cross. The book of Acts speaks of "the church of God, which he hath purchased with his own blood" (Acts 20:28). Cyril of Alexandria's teaching won at the Council of Ephesus in 431. He condemned anyone who does not confess that the Word of God suffered, was crucified and tasted death in the flesh. Charles Wesley's hymn "And Can It Be" (1738) is still sung today. It speaks of Christ's death as the death of the Immortal, and calls it a mystery.
“Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which he hath purchased with his own blood.”
Cyril condemns anyone who does not confess that the Word of God suffered in the flesh, was crucified in the flesh and tasted death in the flesh. Cyril's teaching prevailed at the Council of Ephesus in 431, which deposed Nestorius, bishop of Constantinople, for dividing Christ into two.
A hymn by Charles Wesley, first published in 1738 and still sung in many churches. It speaks of Christ's death on the cross as the death of the Immortal, and calls it a mystery that even the angels cannot fathom.
The usual answer: two natures
Most churches explain it with the teaching of two natures in one person, defined at the Council of Chalcedon in 451. The Catholic Catechism says that Christ's death was a real death: "It is truly the Son of God made man who died and was buried". At death his soul and body were separated, but his divine person kept hold of both. So, they say, the Son died in his human nature, while his divine nature did not die.
On Christ in the tomb: Christ's death was a real death, which ended his earthly human existence (627), and "It is truly the Son of God made man who died and was buried" (629). At death his soul and body were separated, but his divine person continued to possess both (626, 630).
The council confessed one and the same Christ in two natures, divine and human, and declared him "not parted or divided into two persons, but one and the same Son".
The Bible says God (﷾) cannot die
The Bible's own words close the door. Paul calls God "the King eternal, immortal, invisible" (1 Timothy 1:17), and the One "who only hath immortality" (1 Timothy 6:16). God (﷾) says "I am the Lord, I change not" (Malachi 3:6), and "God is not a man" (Numbers 23:19). Isaiah says the everlasting God (﷾) "fainteth not, neither is weary" (Isaiah 40:28). A God who cannot faint or change cannot die.
“Now unto the King eternal, immortal, invisible, the only wise God, be honour and glory for ever and ever. Amen.”
“Who only hath immortality, dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting. Amen.”
“For I am the Lord, I change not; therefore ye sons of Jacob are not consumed.”
“God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?”
“Hast thou not known? hast thou not heard, that the everlasting God, the Lord, the Creator of the ends of the earth, fainteth not, neither is weary? there is no searching of his understanding.”
Not which nature, but who
A nature does not die; a person does. The Council of Chalcedon says there is only one person in Christ, God the Son. So if that person died, God (﷾) died, and the Bible says God (﷾) alone has immortality. If instead the one who died was not God, then only a man died, and God (﷾) did not die for anyone. On the cross Jesus cried, "My God, my God, why hast thou forsaken me?" (Matthew 27:46). That is a man calling on his God. The page on whether Jesus has two natures takes this further.
The council confessed one and the same Christ in two natures, divine and human, and declared him "not parted or divided into two persons, but one and the same Son".
“Who only hath immortality, dwelling in the light which no man can approach unto; whom no man hath seen, nor can see: to whom be honour and power everlasting. Amen.”
“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?”
Allah (﷾), the Ever-Living
In the Qur'an, God (﷾) is "the Living, the Self-subsisting, Eternal. No slumber can seize Him nor sleep" (2:255). He is the One "Who lives and dies not" (25:58), and "Everything (that exists) will perish except His own Face" (28:88). The Prophet (ﷺ) used to pray to Allah (﷾) as the One who does not die, while jinn and human beings die (Bukhari 7383). When the Prophet (ﷺ) himself died, Abu Bakr told the grieving people: whoever worshipped Muhammad, Muhammad is dead; whoever worshipped Allah (﷾), Allah (﷾) is alive and never dies (Bukhari 3668).
“Allah! There is no god but He,-the Living, the Self-subsisting, Eternal. No slumber can seize Him nor sleep. His are all things in the heavens and on earth. Who is there can intercede in His presence except as He permitteth? He knoweth what (appeareth to His creatures as) before or after or behind them. Nor shall they compass aught of His knowledge except as He willeth. His Throne doth extend over the heavens and the earth, and He feeleth no fatigue in guarding and preserving them for He is the Most High, the Supreme (in glory).”
“And put thy trust in Him Who lives and dies not; and celebrate his praise; and enough is He to be acquainted with the faults of His servants;-”
“And call not, besides Allah, on another god. There is no god but He. Everything (that exists) will perish except His own Face. To Him belongs the Command, and to Him will ye (all) be brought back.”
Ibn Abbas reports that the Prophet (ﷺ) used to say: I seek refuge in Your might; there is no god but You, who does not die, while the jinn and human beings die.
Aisha reports that when the Prophet (ﷺ) died, Umar insisted he was not dead. Abu Bakr went in, uncovered his face and kissed him, then came out and told the people: whoever worshipped Muhammad, Muhammad is dead; whoever worshipped Allah, Allah is alive and never dies. He then recited Qur'an 39:30 and 3:144.
Jesus: a messenger, and not killed
The Qur'an says that "Christ the son of Mary was no more than a messenger". He and his mother "had both to eat their (daily) food" (5:75), as every human being does. It asks who could stop Allah (﷾) "if His will were to destroy Christ the son of Mary, his mother, and all every - one that is on the earth" (5:17). And it says of Jesus: "they killed him not, nor crucified him" (4:157). Allah (﷾) raised him up to Himself (4:158).
“Christ the son of Mary was no more than a messenger; many were the messengers that passed away before him. His mother was a woman of truth. They had both to eat their (daily) food. See how Allah doth make His signs clear to them; yet see in what ways they are deluded away from the truth!”
“In blasphemy indeed are those that say that Allah is Christ the son of Mary. Say: "Who then hath the least power against Allah, if His will were to destroy Christ the son of Mary, his mother, and all every - one that is on the earth? For to Allah belongeth the dominion of the heavens and the earth, and all that is between. He createth what He pleaseth. For Allah hath power over all things."”
“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;-”
Perspectives
The Son died in his human nature
Most Christians hold that Jesus is God the Son, one person in two natures. He truly died on the cross in his human nature, they say, while his divine nature did not die.
God the Son died on the cross: the church was bought with God's own blood.
Rebuttal
That would mean that God (﷾) died. Yet Paul calls God the One "who only hath immortality" (1 Timothy 6:16), and the Qur'an calls Him the One "Who lives and dies not" (25:58).
Only his human nature died; his divine nature did not die.
Rebuttal
A nature does not die; a person does. And Chalcedon says the one person in Christ is God the Son. If the one who died was not God, then God (﷾) did not die for anyone, and a man died. On the cross Jesus cried, "My God, my God, why hast thou forsaken me?" (Matthew 27:46): a man calling on his God.
Only his body died; God did not die.
Rebuttal
When a man dies, we do not say his body died; we say he died. The Catechism itself says "It is truly the Son of God made man who died and was buried". Death is when the body stops working and the soul leaves it. If only a body died, God (﷾) did not die for anyone. If the one who died was God, then God (﷾) lost all life in his body, yet the Bible says he "shall neither slumber nor sleep" (Psalm 121:4), and the Qur'an says "No slumber can seize Him nor sleep" (2:255).
The Islamic perspective
God (﷾) is the Ever-Living, who never dies. Jesus was His messenger, a man who ate food like other men; the Qur'an says he was not killed, but raised up to God (﷾).
Allah (﷾) is the Living, who never dies, and everything else perishes.
When the Prophet (ﷺ) died, Abu Bakr said: whoever worshipped Allah (﷾), Allah (﷾) is alive and never dies.
Jesus was a messenger who ate food, and the Qur'an says he was not killed or crucified.