(I was recently asked about this issue, and the following are some notes I put together about it.)
I. Fr. James Edmund O’Reilly, The Relations of the Church to Society – Theological Essays (1882)
If we inquire how ecclesiastical jurisdiction has been continued, the answer is that it in part came and comes immediately from God on the fulfilment of certain conditions regarding the persons. Priests having jurisdiction derive it from bishops or the pope. The pope has it immediately from God, on his legitimate election. The legitimacy of his election depends on the observance of the rules established by previous popes regarding such election. …
We may here stop to inquire what is to be said of the position, at that time, of the three claimants, and their rights with regard to the Papacy. In the first place, there was all through, from the death of Gregory XI in 1378, a Pope – with the exception, of course, of the intervals between deaths and elections to fill up the vacancies thereby created. There was, I say, at every given time a Pope, really invested with the dignity of the Vicar of Christ and Head of the Church, whatever opinions might exist among many as to his genuineness; not that an interregnum covering the whole period would have been impossible or inconsistent with the promises of Christ, for this is by no means manifest, but that, as a matter of fact, there was not such an interregnum. …
The great schism of the West suggests to me a reflection which I take the liberty of expressing here. If this schism had not occurred, the hypothesis of such a thing happening would appear to many chimerical. They would say it could not be; God would not permit the Church to come into so unhappy a situation. Heresies might spring up and spread and last painfully long, through the fault and to the perdition of their authors and abettors, to the great distress too of the faithful, increased by actual persecution in many places where the heretics were dominant. But that the true Church should remain between thirty and forty years without a thoroughly ascertained Head, and representative of Christ on earth, this would not be. Yet it has been; and we have no guarantee that it will not be again, though we may fervently hope otherwise. What I would infer is, that we must not be too ready to pronounce on what God may permit. We know with absolute certainty that He will fulfill His promises. We may also trust that He will do a great deal more than what He has bound Himself by his promises. We may look forward with cheering probability to exemption for the future from some of the trouble and misfortunes that have befallen in the past. But we, or our successors in the future generations of Christians, shall perhaps see stranger evils than have yet been experienced, even before the immediate approach of that great winding up of all things on earth that will precede the day of judgment. I am not setting up for a prophet, nor pretending to see unhappy wonders, of which I have no knowledge whatever. All I mean to convey is that contingencies regarding the Church, not excluded by the Divine promises, cannot be regarded as practically impossible, just because they would be terrible and distressing in a very high degree.
II. A. Dorsch, Institutiones Theologiae Fundamentalis (1928)
The Church therefore is a society that is essentially monarchical. But this does not prevent the Church, for a short time after the death of a pope, or even for many years, from remaining deprived of her head. Her monarchical form also remains intact in this state.
III. Sylvester Berry, The Church of Christ (Eugene, OR: Wipf and Stock, 2009 ), pp. 94-95
OBJECTION I. At the time of the Western Schism the Catholic Church lost her unity for many years by being divided into two, and even three, parties each following a pope of its own choosing.
ANSWER.—The Western Schism caused great harm to the Church in many ways, but it did not affect her unity. After the death of Gregory XI, in 1378, the cardinals proceeded to elect Urban VI as his successor. Three months later, several cardinals claimed the election of Urban to be invalid and selected Robert of Geneva as Pope, under the name of Clement VII. Differences of opinion naturally arose regarding the validity of these elections; some believed Urban VI the rightful pope, while others accepted Clement VII. In 1409 an attempt was made to remedy this situation, but the result was disappointing, and matters were made worse by the election of a third claimant, who took the name Alexander V. Thus matters continued until the Council of Constance, in 1417, when Martin V was elected and recognized by all as the lawful Pope.
At no time during these troubles did any one ever entertain the idea that there were three popes, or that the Church was divided in its government. All admitted that there could be but one legitimate pope, and each party followed the one whom they believed to be lawfully elected successor of St. Peter. The Church was no more divided by the schism than our own government would be by a disputed election to the office of presidency.
IV. Msgr. G[?]. Van Noort, Dogmatic Theology, Volume II: Christ’s Church (Cork, Ireland: Mercier Press, 1958), p. 131
It might seem that unity of rule suffered a setback in the Church at the time of the Western Schims, when for forty years (1378-1417) two or three men claimed to be sovereign pontiff. But with the preservation of unity of faith and communion, hierarchical unity was only materailly, not formally, interrupted. [The same occurs during any papal interregnum, since, from the death of one pope to the election of the next, Catholics literally have no idea who is the Roman Pontiff. Yet, despite this material lack of the ruling power, Catholics are still visibly and formally united under the same principle of authority. — EBB] Although Catholics were split three ways in their allegiance because of the doubt as to which of the contenders had been legitimately elected, still all were agreed in believing that allegiance was owed the one legitimate successor of Peter, and they stood willing to give that allegiance. Consequently, those who through no fault of their own gave their allegiance to an illegitmate pope would no more be schismatics than a person would be a heretic who, desirous of following the preaching of the Church, would admit a false doctrine because he was under the impression that it was taught by the Church.
V. Frs. Rumble and Carty, Radio Replies, Volume 3 (1942)
413. … Although there were three rival claimants to the office of Pope at that time, each with his own following, it is clear that, as a matter of fact, Catholics were not subject to one single Pope. But, as a matter of law, they were. And that makes all the difference. No Catholic said that there ought to be three Popes. All admitted that there should be only one, and that only one of the three could be the lawful Pope. There was, then a lawful Pope, however confused people may have been as to which one was the lawful Pope. The office for which the claimants were contending was the office of St. Peter. And it was to this office that the authority of St. Peter was annexed. In the civil order, no one will admit that the authority attached to the throne in some given kingdom is lost because some pretender wins the allegiance of a certain number of subjects. And when the pretender dies, or renounces his claim, and the subjects revert to a single king whom all acknowledge as lawful ruler, no one holds that his authority is due to the return of the subjects who were deceived. The authority all along was inherent in his office. So the authority annexed to the office of the Pope persisted continuously and in unbroken descent from St. Peter. Subsequent endorsement of that authority by parties who had been led astray did not confer that authority, but merely acknowledged it as possessed by one particular Pope to the exclusion of all pretenders.
VI. Frs. Rumble and Carty, Radio Replies, Volume 4 (1954)
152. Did not the Great Western Schism disprove the unity of the Catholic Church?
No. What is known as the “Great Western Schism” lasted from 1378 till 1417. It was not strictly a schism at all. During that period besides the lawful Pope, there were two others who unlawfully proclaimed themselves Popes, each having a following convinced that his claims were right. By 1417 things were straightened out, and one lawful Pope was acknowledged by all. The unity of the Church as a Church was no more affected by this than the unity of the Kingdom of England was affected by the fact that at various times there were Pretenders to the Throne with followers convinced of their rightful claims.
153. Where was unity then?
However confused Catholics might have been as to which was the true Pope, all admitted the truth of the one Catholic Church, all acknowledged Papal authority, and all agreed that there could be but one true Pope. They held no schismatical principles, and were not schismatics.
154. How do you explain such confusion?
It is explained by the liability of the human judgment to arrive at wrong conclusions, above all when people are not familiar with all the facts of the case. In quite good faith people can support wrong causes. The essential unity of the Catholic Church is not affected by the fact that all Catholics are not individually infallible in such matters.
155. The Great Western Schism must have permanently injured the Church.
It did not. It was but a transitory historical episode; and as I have said was not really a schism at all. The confusion was ended by the Council of Constance in 1417 when the lawful Pope Gregory XII resigned, the two Pretenders were declared not lawful Popes at all, and Pope Martin V was elected and acknowledged by everybody as the one true Pope. Over five hundred years have elapsed since then without any recurrence of such confusion.