Short answer
No. Psalm 16 is a prayer of David, and it opens with "Preserve me, O God". David trusts God (﷾) not to abandon his soul to Sheol, the realm of the dead. That is a prayer to be kept from death, not a promise of rising after it. In Psalm 30 David uses the same word, Sheol, for his own rescue. God (﷾) brought his soul up from it and kept him alive. Acts reads the last word of verse 10 as "corruption", the decay of a body, following the old Greek translation. The Hebrew word, shachat, usually means the pit, and the Jewish translation renders it so. No verse in the Hebrew Bible says that a prophet will die for sins and rise on the third day. Islam's answer is that the psalm fits a servant whom God (﷾) keeps from death.
Full explanation
What the psalm says in its own words
The psalm is headed as David's, and its first words ask God (﷾) to preserve him. The Jewish Publication Society translation is a standard Jewish English Bible. In it, the speaker's flesh dwells in safety. God (﷾) will not abandon his soul to the nether-world, which is Sheol, the realm of the dead. Nor will He let His godly one see the pit. Sheol is not a grave in the ground. The psalm is a man's trust that God (﷾) will keep him alive.
“Michtam of David. Preserve me, O God: for in thee do I put my trust.”
“I have set the LORD always before me; Surely He is at my right hand, I shall not be moved. Therefore my heart is glad, and my glory rejoiceth; my flesh also dwelleth in safety; For Thou wilt not abandon my soul to the nether-world; Neither wilt Thou suffer Thy godly one to see the pit. Thou makest me to know the path of life; In Thy presence is fulness of joy, In Thy right hand bliss for evermore.”
How Acts reads the psalm
In Acts, Peter, a disciple of Jesus, argues that David died and was buried, and his tomb was still there. So, he says, David was speaking as a prophet about the Messiah, the promised king. In this reading the Messiah's soul was not left in hell, and his flesh did not see corruption. Paul, a Christian missionary of the first century, makes the same argument in Acts 13.
“Men and brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulchre is with us unto this day. Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption.”
“Wherefore he saith also in another psalm, Thou shalt not suffer thine Holy One to see corruption. For David, after he had served his own generation by the will of God, fell on sleep, and was laid unto his fathers, and saw corruption: But he, whom God raised again, saw no corruption.”
“For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption.”
David uses the same words of himself
In Psalm 30 David thanks God (﷾) for bringing his soul up from the nether-world, the same word Sheol. He thanks God (﷾) for keeping him alive. In the same psalm he asks what profit there would be if he went down to the pit. That is the same word, shachat, as in Psalm 16. Both words describe a living man rescued from death. In Psalm 86 he says God (﷾) has delivered his soul from the lowest hell. David was alive when he wrote both.
“O LORD, Thou broughtest up my soul from the nether-world;”
“What profit is there in my blood, when I go down to the pit? Shall the dust praise Thee? shall it declare Thy truth?”
“For great is thy mercy toward me: and thou hast delivered my soul from the lowest hell.”
"Corruption" and "Holy One"
The Hebrew word at the end of Psalm 16:10 is shachat, which most often means the pit. Acts follows the old Greek translation of the psalm, which renders it "corruption". Peter's argument leans on that Greek word. The King James Version prints the Hebrew word hasid as "Holy One". It means a godly or faithful man. The same word is used of the godly man in Psalm 4, whom God (﷾) sets apart for Himself.
“I have set the LORD always before me; Surely He is at my right hand, I shall not be moved. Therefore my heart is glad, and my glory rejoiceth; my flesh also dwelleth in safety; For Thou wilt not abandon my soul to the nether-world; Neither wilt Thou suffer Thy godly one to see the pit. Thou makest me to know the path of life; In Thy presence is fulness of joy, In Thy right hand bliss for evermore.”
“For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption.”
“But know that the Lord hath set apart him that is godly for himself: the Lord will hear when I call unto him.”
No third-day prophecy for sins
Paul says that Christ died for our sins and rose on the third day "according to the scriptures". Yet no verse in the Hebrew Bible says that a prophet will die for others' sins and rise on the third day. Isaiah 53, which Christians cite, speaks of an unnamed servant. It says nothing of a rising on the third day. The Law says plainly that every man shall be put to death for his own sin, as Ezekiel repeats. Jesus himself said that every man will be rewarded according to his works. The Qur'an states that no bearer of burdens bears the burden of another.
“For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures:”
“The fathers shall not be put to death for the children, neither shall the children be put to death for the fathers: every man shall be put to death for his own sin.”
“The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.”
“For the Son of man shall come in the glory of his Father with his angels; and then he shall reward every man according to his works.”
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
If the psalm is read of Jesus
Read as it stands, the psalm describes a servant whom God (﷾) keeps from death. Hebrews says that Jesus prayed to the One able to save him from death, and was heard. A prayer to be saved from death, when heard, is answered by rescue. The Qur'an states that his enemies did not kill him or crucify him, and that Allah (﷾) raised him up to Himself.
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
“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;-”
Verse by verse
Acts 2:29-31 — Acts of the Apostles, chapter 2, verses 29-31
“Men and brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulchre is with us unto this day. Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption.”
Cited for
David died and his tomb is still here, so the promise that God's Holy One would not see corruption cannot be about David. It must be about Christ's resurrection.
The response
The argument leans on the Greek word "corruption". The Hebrew says God (﷾) will not let His godly one see the pit. David uses those same words, Sheol and the pit, of his own rescue in Psalm 30. The psalm asks to be spared death. It does not describe dying and rising. Like Psalm 30, it is the prayer of a living man who trusts God (﷾) to keep him from death. If it is read of Jesus, it describes a man God (﷾) kept from death, which is what the Qur'an teaches.
Where it lands — Psalm 16 is David's trust that God (﷾) will keep him from death, so it is no prophecy of a resurrection.
Hosea 6:1-2 — Nevi'im, Hoshea, chapter 6, verses 1-2
“Come, and let us return unto the Lord: for he hath torn, and he will heal us; he hath smitten, and he will bind us up. After two days will he revive us: in the third day he will raise us up, and we shall live in his sight.”
Cited for
Hosea's "in the third day he will raise us up" is the scripture behind the third-day resurrection.
The response
Hosea was a prophet of the eighth century BC. His verse speaks of "us", and it follows a call to the people: "Come, and let us return unto the Lord". It is about the nation of Israel turning back to God (﷾) and being healed. It names no single man, no death for sins and no tomb.
Where it lands — Hosea speaks of the people's return to God (﷾), not of one man rising from the dead.
Perspectives
Mainstream Christian
Christians say David died and his body decayed, so Psalm 16 cannot be about him. They say he spoke as a prophet of the Messiah, whom God raised from the dead before his body could decay.
Peter in Acts 2 argues that David died and was buried, so the psalm speaks of the resurrection of Christ.
Rebuttal
Peter's argument leans on the Greek word "corruption", but the Hebrew says God (﷾) will not let His godly one see the pit. David uses those same words, Sheol and the pit, of his own rescue in Psalm 30. That David died years later does not turn his prayer for rescue into a promise of rising.
Paul in Acts 13 argues that David saw corruption, but the one God raised saw none.
Rebuttal
Paul repeats the same reading, and it leans on the Greek "corruption". The Jewish translation renders the Hebrew shachat as "the pit". And no verse in the Hebrew Bible says that a prophet will die for sins and rise on the third day.
The Islamic perspective
Psalm 16 is David's trust that God (﷾) will keep him from death. It says nothing of a death for sins or a rising on the third day. If it is applied to Jesus at all, it fits a servant whom God (﷾) saved from death, as the Qur'an teaches.
The psalm is headed as David's and opens with a prayer that God (﷾) preserve him.
Sheol is the realm of the dead, and not being abandoned to it means being kept from death.
David uses the same two words, Sheol and shachat (the pit), of his own danger and rescue while alive.
Acts reads a Hebrew word as "corruption". The Jewish translation renders it "the pit", and Acts follows the Greek wording.
No verse in the Hebrew Bible says a prophet will die for others' sins and rise on the third day. The Law says every man dies for his own sin.
The Qur'an teaches that God (﷾) saved Jesus. He was neither killed nor crucified, but raised by Allah (﷾) to Himself.