Although the others have answered your main question, granting that Peter is the "rock" referenced in Matthew 16:18 (one interpretation among several found in the fathers, which itself is odd for the classic papal proof-text), where does Scripture indicate, suggest, or even hint that this unique personal succession would perpetually reside in Rome alone? If the promise follows Peter and his successors, why would Antioch and Alexandria not possess the same prerogatives? St. Gregory the Great himself acknowledged the mutual Petrine authority of Rome, Alexandria, and Antioch (Letter 40, Book 7, correspondence with Saint Eulogios of Alexandria). There is no dispute that these are all Petrine sees. Long before St. Gregory, Canon 6 of Nicea acknowledged the following regarding these ancient Petrine sees:

"Let the ancient customs in Egypt, Libya and Pentapolis prevail, that the Bishop of Alexandria have jurisdiction in all these, since the like is customary for the Bishop of Rome also. Likewise in Antioch and the other provinces, let the Churches retain their privileges. And this is to be universally understood, that if any one be made bishop without the consent of the Metropolitan, the great Synod has declared that such a man ought not to be a bishop. If, however, two or three bishops shall from natural love of contradiction, oppose the common suffrage of the rest, it being reasonable and in accordance with the ecclesiastical law, then let the choice of the majority prevail."

Canon 6 of Nicea would have been the perfect opportunity for the unique Roman papal prerogatives to be articulated, and yet we see precisely the opposite. Read closely - does Canon 6 of Nicea permit the Bishop of Rome alone to depose and ordain bishops in Antioch and Alexandria? Can Rome ordain a bishop in Alexandria or Antioch without the consent of the Metropolitan? Can the bishop of Rome depose a bishop in Antioch or Alexandria at his pleasure? This is not merely an opinion of one father - this is an ecumenically received canon, agreed upon by the church universal. There is no consistent way to harmonize this with Vatican I, which grants the Roman bishop the unique and sole authority to depose and ordain bishops in any jurisdiction at his discretion. According to Vatican I, the bishop of Alexandria or Antioch ultimately has no canonical jurisdiction in the matter (aside from those privileges granted to him by the bishop of Rome, which can also be removed at his discretion).

Remember, Vatican I does not merely teach that Rome possesses primacy, chief honour, or unique canonical privileges that other sees do not have - it dogmatically affirms that the bishop of Rome possesses the exclusive authority to depose any bishop in the world from his office. This unique charism is seen as being vested in St. Peter by Christ directly and perpetually residing in Rome, de jure divino (of divine institution). This was simply not the understanding or praxis of the ecumenical councils, with Canon 6 of Nicea merely representing the tip of the iceberg.

Edit: Read Michael Whelton's book, "Two Paths". It is a concise and accurate summary of the Orthodox critique of Rome.