Now I know this is a complicated subject.
It would seem that Christ did give Peter a primacy and it would seem that the Bishop of Rome was seen as being the successor to this primacy and of St. Paul.
Now it also seems that the early conception of the primacy of the Bishop of Rome in no way resembles the infallible, universal bishop we now see. But there are some interesting patristic instances where we see the Pope having what some may say are extraordinary powers.
The main instance I can think of is when Pope St. Leo intervened in the Council of Chalcedon and vetoed Canon 35. However, I must mention a first contradiction to modern Catholic doctrine in the instance; Letter XXXVII presents how St. Leo felt that a council was unnecessary, but that he "[has] bestowed [his] zeal upon obeying your clemency's commands." So much for Vatican I's assertion that only a Pope can convene a general council.
But in Letter CV:III he says about Canon 35, "But the bishops' assents, which are opposed to the regulations of the holy canons composed at Nicaea in conjunction with your faithful Grace, we do not recognize, and by the blessed Apostle Peter's authority we absolutely dis-annul in comprehensive terms, in all ecclesiastical cases obeying those laws which the Holy Ghost set forth by the 318 bishops for the pacific observance of all priests in such sort that even if a much greater number were to pass a different decree to theirs, whatever was opposed to their constitution would have to be held in no respect."
The original Latin is (PL 54):
Consensiones vero episcoporum, sanctorum canonum apud Nicaeam conditorum regulis repugnantes, unita nobiscum vestrae fidei pietate in irritum mittimus, et per auctoritatem beati Petri apostoli, generali prorsus definitione cassamus, in omnibus ecclesiasticis causis his legibus obsequentes quas ad pacificam observantiam omnium sacerdotum, per trecentos decem et octo antistites Spiritus sanctus instituit: ita ut etiam si multo plures aliud quam illi statuere decernant, in nulla reverentia sit habendum, quidquid fuerit a praedictorum constitutione diversum.
However, while on the one hand this serves the Roman cause, the reasons St. Leo gives for this has nothing to do with the universal nature of the Roman Pontiff, but a read through Letter CV will show that St. Leo mainly does not want this because Constantinople is not really of apostolic rank and because it would be violating the canons of other Councils. Leo himself says that "The Nicene canons are unalterable and binding universally." A modern Catholic would see this reasoning as faulty and plain wrong and I think we can conclude that the reasons for Leo's rejection of this canon has to do with an argument built on sand (Text of Letter Found Here: http://www.newadvent.org/fathers/3604105.htm
A quick read of Letter CVI reveals Leo's opinion on this matter: "These holy and venerable fathers who in he city of Nicaea, after condemning the blasphemous Arius with his impiety, laid down a code of canons for the Church to last till the end of the world, survive not only with us but with the whole of mankind in their constitutions; and, if anywhere men venture upon what is contrary to their decrees, it is ipso facto null and void..."
The real question, however, is whether or not St. Leo actually had the authority to cassate a decree and if there are any examples of any other Popes or Patriarchs doing this. It would seem that St. Leo is cassating it because he thinks that it is "ipso facto null and void," but he also uses the authority of Peter to cassare
(to null, to void, to annull; used in a legal sense) the canon. It's hard to say which perspective is correct.
The East, however, came to accept Canon 35 of Chalecedon and it appeared in the Western copies eventually too. Does this mean it was still approved by the Council? I think someone with a knowledge of the Acts of the Council of Chalcedon might be able to tell us.
Nevertheless, this brings me to a second question. When the West introduced new doctrines, thereby schismating themselves from the East, did the primacy of Peter move from the See at Rome to the See at Constantinople? Are there any Fathers who recognized this as happening?
PS. St. Leo also claims that the power of Peter passed to all Bishops, so it's hard to see where he stands.