No. Christ was. He is the Stone as you say. Yet Peter is His vicar. He was Christ's human substitute on earth after Jesus' ascension. He was left in charge with remarkably similar authority to what Christ Himself had, though, of course, it was delegated to him. Jesus had it by divine right as part of the Godhead, Peter merely had it by delegation. Let's be clear here.
Just the same, delegated though it may be, to emphasize and ratify what He knew He would do with Peter, Jesus attributed to him in Matthew 16 a name which is entirely evocative of the title Jesus Himself has eternally by divine right. If anything, there is much to ponder in the simple fact of what Christ did in choosing Kepha as Peter's new name. He highlights Peter's new authority as nearly identical with his own.
Rule by committee is never effective in a large organization, as anyone familiar with human nature would attest. Jesus is far more familiar with human nature than any of us will ever be ourselves, and He knew the limitations of committees. Peter's authority, like Abraham's, like Moses', like Joshua's, like David's and Solomon's, was invested in a sole personage. Unlike the others, though, Peter did not merely pass-on edicts and instructions from God, he ACTUALLY was Christ's vicar, who *commanded* in His stead. His successors have done the same, to this very day. The Spirit guides them in this, and safeguards their teaching.
Other than the somewhat ambiguous "rock" statement I don't suppose you have any scripture to back that up, do you?
My reading of scripture suggests that if there ever was a "pope" of Rome in the first century, that it must have been Paul, who specifically claimed to be the Apostle to the Romans and to whose authority Peter deferred.
Frankly, I really don't think there was a first "Pope" as Christ told his disciples clearly to call no man "Father" and he also indicated that no one is "Holy" except for God. So calling any human being a "Holy Father" directly contradicts the clear teaching of Christ on the subject.
And where exactly do you get this "Vicar of Christ" idea, anyway? Seems to me that whatever purpose a "Vicar of Christ" could perform was fulfilled in the sending of the Holy Spirit as teacher and comforter. The "Pope" therefore seems to have usurped the Authority of the Holy Spirit.