The Fathers on the Necessity of Communion with Rome

"All the churches of God salute you."
Romans 16:16b

The Papacy, as it functions in the modern day, is a development. In the first millennium, its role can be compared to a Supreme Court, whereas in the second millenium it behaves more like a CEO. But, true to its name, the Papacy has always been positioned as a father, and as a rock to the whole church.

This means the Papacy’s outward framework, the pragmatic method by which it operates, has evolved as time has gone on, but that its interior animating principle, its theological purpose, is consistent with what came before it and is in harmony with its past.

The early Fathers may not utter in any explicit fashion such things that the Pope is infallible when he speaks ex cathedra, or that he holds universal and immediate jurisdiction over the whole church. But they do refer to his see as having always maintained the apostolic faith and never suffering its corruption, and that it is preeminent among and head over all the churches. They admonish that there is no unity in the Church except through unity with Rome. They record in that no theological dispute can be counted as settled, nor universal church law set into motion, without his approval.

The Pope's universal primacy over the Church serves to maintain and manifest the universal unity of faith (Eph 4:13) and bond of peace (4:3). He is the direct and preeminent successor of St. Peter, chief and head of the Apostles, the rock upon whom Christ built his Church (Mt 16), whom He commissioned to feed His sheep (Jn 21), for whom He prayed for unfailing faith and indicated that he was to strengthen the brethren (Lk 22.) Thus, the role Peter serves as established by Christ, is the one which the Pope and the Church of Rome continue to exercise.

These excerpts from the Fathers will highlight a common theme which adds weight to the papal claims.

St. Irenaeus of Lyons (c. AD 130 - 200)

Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who... assemble in unauthorized meetings by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul...  For it is a matter of necessity that every Church should agree with this Church, on account of its pre-eminent authority, that is, the faithful everywhere, inasmuch as the apostolical tradition has been preserved continuously by those who exist everywhere.
(Against Heresies 3:3:2)

Every church should agree with the church of Rome "on account of its preeminent authority", inasmuch as that church has continuously preserved the apostolic tradition.

St. Cyprian of Carthage (c. AD 210 - 258)

After such things as these, moreover, they still dare — a false bishop having been appointed for them by, heretics— to set sail and to bear letters from schismatic and profane persons to the throne of Peter, and to the chief church whence priestly unity takes its source; and not to consider that these were the Romans whose faith was praised in the preaching of the apostle, to whom faithlessness could have no access.
(Letter 58:14)

It is from the Roman Church "whence priestly unity takes its source"

St. Optatus of Milevis (c. AD 397)

You cannot then deny that you do know that upon Peter first in the City of Rome was bestowed the Episcopal Chair [Cathedra], on which sat Peter, the Head of all the Apostles (for which reason he was called Cephas), that, in this one Chair, unity should be preserved by all, lest the other Apostles might claim--each for himself--separate Chairs, so that he who should set up a second Chair against the unique Chair would already be a schismatic and a sinner.
(On the Schism of the Donatists 2:2)

The Chair of Peter is the chair against which all other chairs are held, "in this one Chair, unity should be preserved by all," "he who should set up a second Chair against the unique Chair would already be a schismatic and a sinner."

St. Ambrose of Milan (c. 340 - 397)

Your grace must be besought not to permit any disturbance of the Roman Church, the head of the whole Roman world and of the most holy faith of the Apostles, for from thence flow out to all [churches] the bonds of sacred communion.
(Letter 11:4)

It is from the Roman Church "from thence flows out of all the bonds of sacred communion." It is "the head of... the most holy faith of the Apostles."

St. Jerome (AD 347-420)

My words are spoken to the successor of the fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built! (Matthew 16:18) This is the house where alone the paschal lamb can be rightly eaten. (Exodus 12:22) This is the Ark of Noah, and he who is not found in it shall perish when the flood prevails. (Genesis 7:23)
(Letter 15:2)

As he follows Christ, so he communes with the pope.

St. Maximus the Confessor*

For the extremities of the earth and all in every part of it who purely and rightly confess the Lord, look directly towards the most holy Roman Church and its confession and faith, as it were to a sun of unfailing light, awaiting from it the bright radiance of the sacred dogmas of our fathers, according to what the six inspired and holy councils have purely and piously decreed, declaring most expressly the symbol of faith. For from the coming down of the Incarnate Word amongst us, all the Churches in every part of the world have possessed that greatest Church alone as their base and foundation, seeing that, according to the promise of Christ our Saviour, the gates of Hell do never prevail against it, that it possesses the keys of a right confession and faith in Him, that it opens the true and only religion to such as approach with piety, and shuts up and locks every heretical mouth that speaks injustice against the truth. 

If the Roman See recognizes Pyrrhus to be not only a reprobate but a heretic, it is certainly plain that everyone who anathematizes those who have rejected Pyrrhus, anathematizes the See of Rome, that is, he anathematizes the Catholic Church. I need hardly add that he excommunicates himself also, if indeed he is in communion with the Roman See and the Catholic Church of God.

St. Theodore the Studite
According to the Fathers, the right hand of fellowship must be exchanged with the Church of Rome if one wishes to be considered to be part of the One, Holy, Catholic and Apostolic Church, which is the Bride and Body of Christ. Of this church, distinct from all others, St. Paul gives the unique farewell "all the churches of God salute you," (Romans 16:16) and the churches of God through the centuries have recognized the import of this salutation.