Pope St. Gregory the Great (+604) on the authority of the Bishop of Rome:
"For the Byzacent primate(4) had been accused on some charge, and the most pious Emperor wished him to be judged by us according to canonical ordinance. But then, on the receipt of ten pounds of gold, Theodorus the magister militum opposed this being done. Yet the most pious Emperor admonished us to commission some one, and do whatever was canonical. But, seeing the contrarieties of men, we have been unwilling to decide this case. Now, moreover, this same primate says something about his own intention. And it is exceedingly doubtful whether he says such things to us sincerely, or in fact because he is being attacked by his fellow-bishops: for, as to his saying that he is subject to the Apostolic See, if any fault is found in bishops, I know not what bishop is not subject to it." (The Great Epistles, Epistle LIX: To John, Bishop of Syracuse).
"For as to what they say about the Church of Constantinople, who can doubt that it is subject to the Apostolic See, as both the most pious lord the emperor and our brother the bishop of that city continually acknowledge? Yet, if this or any other Church has anything that is good, I am prepared in what is good to imitate even my inferiors, while prohibiting them from things unlawful. For he is foolish who thinks himself first in such a way as to scorn to learn whatever good things he may see." (Ibid; XII: To John, Bishop of Syracuse)
"We therefore exhort your Fraternity that, conforming to the custom of your churches, as well with respect to the announcement of Easter, as also if need should compel any of you to travel anywhere for business of your own, you should ask leave of your said metropolitan according to the rule imposed upon you; except that, if (as we hope will not be the case) you should happen to have a case against your said Metropolitan, then those who are in haste on this account to seek the judgment of the Apostolic See have licence to do so, as you know is allowed in the canons by the institution even of the ancient Fathers." (Ibid; VIII: To the Bishops of Sardinia).
EPISTLE LXVIII: TO EUSEBIUS OF THESSALONICA.
Gregory to Eusebius of Thessalonica, Urbicus of Dyrrachium, Andrew of Nicopolis, John of Corinth, John of Prima Justiniana, John of Crete, John of Larissa and Scodra, and many other bishops.
"We are constrained by the care of government which we have undertaken to extend vigilantly the solicitude of our office, and to instruct the minds of our brethren by addresses of admonition, that no wrongful presumption. may avail to deceive the ignorant, nor any dissimulation to excuse those who know. Be it known then to your Fraternity that John, formerly bishop of the city of Constantinople, against God, against the peace of the Church, to the contempt and injury of all priests, exceeded the bounds of modesty and of his own measure, and unlawfully usurped in synod the proud and pestiferous title of oecumenical, that is to say, universal. When our predecessor Pelagius of blessed memory became aware of this, he annulled by a fully valid censure all the proceedings of that same synod, except what had therein been done in the cause of Gregory, bishop of Antioch, of venerable memory; taking him to task with most severe rebuke, and warning him to abstain from that new and temerarious name of superstition; even so as to forbid his deacon to go in procession(2) with him, unless he should amend so great a wickedness. And we, adhering in all respects to the zeal of his rectitude, observe his ordinances, under the protection of God, irrefragably, since it is fitting that he should walk without stumbling along the straight way of his predecessor, whom the tribunal of the eternal Judge awaits for rendering an account of the same place of government. In which matter, lest we should seem to omit anything that pertains to the peace of the Church, we once and again addressed the same most holy John by letter, bidding him relinquish that name of pride, and incline the elation of his heart to the humility which our Master and Lord has taught us. And having found that he paid no regard, we have not desisted, in our desire of concord, from addressing the like admonitions to our most blessed brother and fellow-priest Cyriacus, his successor. But since it is the case, as we see, now that the end of this world is near at hand, that the enemy of the human race has already appeared in his harbingers, so as to have as his precursors, through this title of pride, the very priests who ought to have opposed him by living well and humbly, I exhort and entreat that not one of you ever accept this name, that not one consent to it, that not one write it, that not one admit it wherever it may have been written, or add his subscription to it; but, as becomes ministers of Almighty God, that each keep himself from this kind of poisoned infection, and give no place to the cunning lier-in-wait, since this thing is being done to the injury and rendering asunder of the whole Church, and, as we have said, to the contemning of all of you. For if one, as he supposes, is universal bishop, it remains that you are not bishops.
Furthermore, it has come to our knowledge that your Fraternity has been convened to Constantinople. And although our most pious Emperor allows nothing unlawful to be done there, yet, lest perverse men, taking occasion of your assembly, should seek opportunity of cajoling you in favouring this name of superstition, or should think of holding a synod about some other matter, with the view of introducing it therein by cunning contrivances,--though without the authority and consent of the Apostolic See nothing that might be passed would have any force, nevertheless, before Almighty God I conjure and warn you, that the assent of none of you be obtained by any blandishments, any bribes, any threats whatever; but, having regard to the eternal judgment, acquit ye yourselves salubriously and unanimously in opposition to wrongful aims; and, supported by pastoral constancy and apostolical authority, keep out the robber and the wolf that would rush in, and give no way to him that rages for the tearing of the Church asunder; nor allow, through any cajolery, a synod to be held on this subject, which indeed would not be a legitimate one, nor to be called a synod. We also at the same time admonish you, that if haply nothing should be done with mention of this preposterous name, but a synod be by any chance assembled on another matter, ye be in all respects cautious, circumspect, watchful, and careful, lest anything should therein be decreed against any place or person prejudicially, or unlawfully, or in opposition to the canons. But, if any question arises to be treated with advantage, let the question in hand take such a form that it may not upset any ancient ordinances. Wherefore we once more admonish you before God and His Saints, that you observe all these things with the utmost attention, and with the entire bent of your minds. For if any one, as we do not believe will be the case, should disregard in any part this present writing, let him know that he is segregated from the peace of the blessed Peter, the Prince of the Apostles. Let, then, your Fraternity so act that when the Shepherd of shepherds comes in judgment, you may not be found guilty with respect to the place of government which you have received." (Ibid; LXVIII: To Eusebius of Thessalonica).
Gregory to Constantius, Bishop of Mediolanum.
"My most beloved son, the deacon Boniface, has conveyed to me certain private information through thy Fraternity's letter; namely that three bishops, having sought out rather than found an occasion, have separated themselves from the pious communion of your Fraternity, saying that you have assented to the condemnation of the Three Chapters(1), and have given a security(2). And, indeed, whether there has been any mention made of the Three Chapters in any word or writing whatever thy Fraternity remembers well; although thy Fraternity's predecessor, Laurentius, did send forth a most strict security to the Apostolic See, to which most noble men in legitimate number subscribed; among whom I also, at that time holding the praetorship of the city, likewise subscribed; since after such a schism had taken place about nothing, it was right that the Apostolic See should take heed, with the view of guarding in all respects the unity of the Universal Church in the minds of priests." (Ibid; II: To Constantius, Bishop of Mediolanum).
EPISTLE XXX: TO JOHN, SUBDEACON.
Gregory to John, &c
"Inasmuch as it is manifest that the Apostolic See is, by the ordering of God, set over all Churches, there is, among our manifold cares, especial demand for our attention, when our decision is awaited with a view to the consecration of a bishop. Now on the death of Laurentius, bishop of the church of Mediolanum, the clergy reported to us that they had unanimously agreed in the election of our son Constantius, their deacon. But, their report not having been subscribed, it becomes necessary, that we may omit nothing in the way of caution, for thee to proceed to Genua (Genoa), supported by the authority of this order(2). And, inasmuch as there are many Milanese at present there under stress of barbarian ferocity, thou must call them together, and enquire into their wishes in common. And, if no diversity of opinion separates them from the unanimity of the election--that is to say, if thou ascertainest that the desire and consent of all continues in favour of our aforesaid son, Constantius,--then thou art to cause him to be consecrated by his own bishops, as ancient usage requires, with the assent of our authority, and the help of the Lord; to the end that through the observance of such custom both the Apostolic See may retain the power belonging to it, and at the same time may not diminish the rights which it has conceded to others." (Ibid; XXX: To John, Subdeacon).
"Furthermore, I inform you that I have received a letter from the most pious Lord desiring me to be pacific towards my brother and fellow-priest John. And indeed so it became the religious Lord to give injunctions to priests. But, when this my brother with new presumption and pride calls himself universal bishop, having caused himself in the time of our predecessor of holy memory to be designated in synod by this so proud a title, though all the acts of that synod were abrogated, being disallowed by the Apostolic See,--the most serene Lord gives me a somewhat distressing intimation, in that he has not rebuked him who is acting proudly, but endeavours to bend me from my purpose, who in this cause of defending the truth of the Gospels and Canons, of humility and rectitude; whereas my aforesaid brother and fellow-priest is acting against evangelical principles and also against the blessed Apostle Peter, and against all the churches, and against the ordinances of the Canons." (Ibid; XX: To Mauricius Augustus).
"Nay, but dost thou not thyself know that the case which arose on the part of the presbyter John against John of Constantinople, our brother and fellow-bishop, came before the Apostolic See, and was decided by our sentence?,(1) If, then, a cause was brought under our cognizance from that city where the prince is, how much more should an affair between you have the truth about it ascertained and be terminated here?" (Ibid; XXIV: To Marinianus, Bishop of Ravenna).
All underlining is mine.
Although it is true that St. Gregory eschewed the title "Universal" (Ecumenical), he did not deny the authority of his office as Bishop of Rome and successor of St. Peter.
That should be apparent from the passages above.