More thoughts that overlap with earlier posts but perhaps a a little to the thoughts.
St Peter was the Prince and leader of the Apostles, there is no doubt about this. Also, there are leading Sees in the Church, which all, except Constantinople and Jerusalem, lay direct claim to being Sees of Peter and rightly so. Antioch and Alexandra, through St Mark, are connected to St Peter because they are Sees of Primacy and hence Sees of Peter. These Sees of Primacy are important for the unity and world wide witness of the Church. The See of Rome is the chief of these Sees and so sums up all of them in one, without excluding the privileges of the other Sees. The comments posted are quite in keeping with Orthodox ecclesiology. Also, the See of Rome, or the chief See of Primacy will not fail, also acceptable and Orthodox. In fact, from an Orthodox perspective, apart from Rome, all the primary Sees are still Orthodox (Catholic) today. From a Roman perspective, it must be said how to explain that all the other Sees have fallen for so long and not only for a few years/decades as in previous cases before the “Schism of 1054?.
Has Rome indeed fallen? For the Orthodox the answer is in a sense yes, the Old Rome fell, but in the main sense no because the See of Rome continues in New Rome, which remains the See of Peter in the Church. New Rome cannot claim to be a See founded by Peter as a See of Primacy because in the apostolic times it was not worthy or suitable for such a place but with Constantine founding a new capital there, it needed to share the Primacy with Rome because it too became the city of the world.
If it was possible for St Paul to become a Prince of the Apostles first equal with St Peter without negating St Peter being the focus of unity and leadership, two Princes in one Prince, then it is possible for New Rome to share equality with Old Rome without negating Old Rome being the focus of unity and leadership. It shares the one leadership of Old Rome, which continues to be named as such even though in practice there are two equal heads.
So, I would argue that the statements posted about Rome can stand well in Orthodoxy without any need to hide them under the carpet. Although, from my reading of the histories of the Councils, it was the Roman Emperor in the East who called the Ecumenical Councils and based them around the effective Capital of the Empire New Rome, the Patriarch of which was quite rightly deserving of the title Ecumenical by the late Sixth/Seventh Century.
What is meant by Petrine primacy? For myself it is the principle of a “single” point of unity and leadership to manifest the unity of the Church. This point can be manifested at various levels from regional Metropolis Sees to Patriarchal Sees and then “universally” in Rome. Because Peter was the Prince of the Apostolic Order, he is the icon or type of this principle and, as in all matters of Christian praxis, icon and types are incarnated in practical reality so Peter physically goes to Rome, because it is the Capital of the Empire, to establish it by his presence rather than merely proclaiming the See from elsewhere because of its position. The Sees of Alexandria and Antioch also have a similar connection to Peter because they have Petrine Primacy in their own Patriarchates. (Note: this principle of unity is different from that of a Bishop within a diocese because it is an order inter-Sees rather than intra-See, so I am not so strong on talking of Petrine Primacy belonging to every Bishop but neither do I exclude it to some degree.) Also, as William Tighe pointed out from St Leo, once the position of primacy becomes established in the Church it is irrevocable; this also applies to the Sees of Alexandria and Antioch. (Jerusalem being a special case). (Note: what is meant by irrevocable needs to be examined but whatever irrevocability is applied to Rome must also be used for Alexandria and Antioch. Thus if irrevocability means never falling away from the Church then this must also be the case for Alexandria and Antioch.)
Now Constantinople is an interesting case. It becomes 300 years after the Apostles the Capital of the Empire. Following Apostolic Tradition it should be a See of Primacy but it also cannot change the established order of the Tradition of the Church. I understand that the solution of the Fathers was to make Constantinople one with Rome as in theory Old Rome and New Rome were one Capital in two locations. Because the name is one it remains with the first and is derived from the first so Constantinople is rightly second in reference to Rome but in the overall order it is first equal with Rome, so not displacing Alexandria as the second in order. When the Fathers speak of having like privileges as Rome it also includes the principle of Petrine Primacy, although historical necessity precluded the presence of Peter initially. (Note: Constantinople traces itself back to St Andrew, the first called and the brother of Peter, to establish Apostolic connection, although I am not sure that this is necessary, although there may be something more in this than I am aware. I am happy with the analogy of St Paul, who was not properly qualified as an Apostle but nevertheless becomes first equal with Peter as Prince of the Apostles, interestingly only Peter’s name is used for the primacy even though both were Princes and fathers of the See in Rome; this ties in with later use of one name for two.) Thus Constantinople is Rome in another location. Petrine Primacy does not “transfer” with the Fall of Rome but is inherently part of Constantinople’s Patriarchy and Primacy. (Nevertheless, in humility the Patriarchs of Constantinople always refer to Old Rome in deference as the “first” See, even though taking the title “Ecumenical”.)
I believe that the issue between Roman Catholics and Orthodox is how to understand Petrine Primacy. The history of Constantinople tends to suggest a monarchal form of Primacy closer to Rome than some may suggest. Nevertheless, I believe that in the East there was a clear sense, even if in practice infringed often, that even Rome or Constantiople could not directly interfere within the jurisdictions of other Sees of Primacy, such as other Patriarchates or even within regions of a Metropolis, even if they appoint the Metropolitan Bishop themselves. This also goes to interference within Dioceses by Metropolitans or Patriarchs. Thus each “level” of headship and unity does not negate the effective headship and unity of each “level” but is rather to harmonise and unite them. So, I think, with a very poor amount of reading on the matter, that the Eastern Patriarchs were not concerned with claims of Primacy as such but with claims of internal jurisdiction within other Patriarchal jurisdictions and thus negating the “real” primacy of these jurisdictions. Even Rome had its own area of Patriarchal jurisdiction, Patriarch of the West, only within which it could exercise that form of control. This understanding, I think, was at least in historical practice the understood relationships of all the Patriarchs including Rome before the Schism (at least as a symbolic demarkation line.)
Nevertheless, Rome, and Constantinople, did/do carry privileges that transcend other Patriarchates and these are those exercised universally, without infringing on the “rights” of Patriarchs because they are in a sense extra-Patriarchal. (I have not really considered the actual privileges apart from the final courts of appeal and focus of mission outside other Patriarchal boundaries. Perhaps those privileges that are not strictly able to be exercised completely within a territorial boundary.)
Because Constantinople is Rome, the belief that Rome would never fail is still true for Orthodox even though Old Rome may have fallen. The principle is maintained even if its original incarnation may have ceased, although there is still an incarnation of the principle. (Similarly the Fathers thought that Rome as secular Empire would never fall until the Antichrist was to appear. The original incarnation of Rome has fallen but the principle can be argued to continue incarnated within other Empires, even carrying the symbols, such of Russia or perhaps the Holy Roman Empire and, although it may not think of itself as such, the United States can be said to maintain the principle today. So, a particular incarnation of a See does not necessarily need to remain for the principle to be maintained but it must nevertheless be validly incarnate somewhere.)
It is amazing that all the Patriarchates are still in practical existence, although maybe only just so. The fact that, from the Orthodox perspective, Rome may have fallen into heresy has not broken any Divine promises. (Perhaps, Divine foresight knew of this and established Constantinople to ensure the continuation.) I think that Roman Catholics have a more difficult job of explaining the fall of Constantinople, Alexandra, Antioch and Jerusalem. One may be ecumenical and say that they have never actually left the Church of Rome, even with all the problems, but I must say that the history of relationships between the Churches does not support that the vast majority of earlier generations acted in this belief, so it must be doubted whether this can really be the case. In some ways the actions of Old Rome tend to see a greater fall than the actions of New Rome. New Rome has never established an Orthodox Bishop of Rome whereas there is a “Latin” Patriarchate in Jerusalem assumes that the Orthodox Patriarch is not truly the “Catholic” Patriarch of Jerusalem.
Whether the Petrine Primacy is with the consciousness of Constantinople as such I am not sure. The principle of the Primacy, I believe, is within the consciousness of Constantinople and perhaps the name, as such, has not been regularly used to avoid confusion with the developed doctrine on the matter in Old Rome. (This also applies to the name Catholic, which is still in the Orthodox consciousness even if not used regularly to avoid confusion.) The reason for my belief is the continued use of the title Ecumenical, which has universal connotations and, if I remember correctly, were behind the objections of St Gregory I of Rome on the use of the name, by St John the Faster of Constantinople. (Excuse me if my historical memory may be a little out here.) This title would only be appropriate if one considered that they had the sort of status that would be appropriately named as Petrine.
Perhaps my thoughts are only theory with no real basis in tradition but I hope that they may have some benefit in trying to explain the evidence available both Western and Eastern.