When our Lord was speaking to Paul [Acts 9:3-9] he never mentioned the word "rock".
Neither was it mentioned by in the book of Mark (8:27-30)after Peter answered "You are the Christ.". No mention of any "rock".
Oh, Luke's account of the same conversation (9:18-20), no "rock" .. only "The Christ of God".
I read these over and over again, and the ONLY common thing in all three gospels, is that JESUS is THE CHRIST .. Son of the living God. Could it be that THAT is the "rock" upon which He built His church: The fact that He is Jesus, Our Christ, Son of the Living God??
That seems to be the theme through the whole NT. One head, Christ Jesus.