Charles Spurgeon Commentary Acts 13:32-37

"And we bring you good tidings of the promise made unto the fathers, that God hath fulfilled the same unto our children, in that he raised up Jesus; as also it is written in the second psalm, Thou art my Son, this day have I begotten thee. And as concerning that he raised him up from the dead, now no more to return to corruption, he hath spoken on this wise, I will give you the holy and sure [blessings] of David. Because he saith also in another [psalm], Thou wilt not give Thy Holy One to see corruption. For David, after he had in his own generation served the counsel of God, fell asleep, and was laid unto his fathers, and saw corruption: but he whom God raised up saw no corruption." — Acts 13:32-37 (ASV)

And we declare unto you glad tidings, how that the promise which was made unto the fathers, God hath fulfilled the same unto us their children, in that he hath raised up Jesus again: as it is also written in the second Psalm, Thou art my Son, this day have I begotten thee. And as concerning that he raised him up from the dead, now no more to return to corruption, he said on this wise, I will give you the sure mercies of David. 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.

So David was not speaking of himself, but he was speaking of another and higher David, his greater Son, the Son of God, begotten of the Father.