III. Valid Elections
However, the claim is made: "We are still in the period of 'sede vacante'; no valid pope has been elected since Pope Pius XII."
To this statement a few things can be said. First, a question: Who has the responsibility of saying that the pope’s election was doubtful? The layman in the street? A Bishop? The College of Cardinals? A Council? There is no clear answer to this question. So, just because someone says an election is invalid, this does not make the election invalid. Have there been elections to the papacy since Pope Pius XII? Yes, there have been 4 elections to the Chair of Peter.
How can we look at these elections? Certain sedevacantists say that they are invalid because the person elected was not a legitimate candidate for the office. For argument’s sake, let us briefly entertain this possibility to show why it in no way would jeopardize the last four pontificates. I defer to the theologian Cardinal Billot, the Doctor St. Alphonsus De Liguori, and the great Benedictine Abbot, Prosper Guéranger. They give the following rule: "The peaceful and universal acceptance of a pope by the whole Church is a sign and effect of a valid election."
Cardinal Billot (the great Jesuit theologian of the first half of this century) states:
"Finally, what one may think of the possibility or the impossibility of an heretical pope, there is at least one point absolutely clear which no one can put in doubt, and it is that the acceptance, the adherence, of the Universal Church to a pope will always be, by itself, the infallible sign of the legitimacy of such-and-such a pontiff; and consequently of the existence of all the conditions required for legitimacy." And this is based on the Church’s attribute of Indefectibility as defined by "the promise of the infallible Providence of Christ [that] ‘the gates of hell shall not prevail against it’ and ‘Behold, I am with you all days even unto the end of the world.’ For the adherence of the Church to a false pontiff would be the same thing as its adherence to a false rule of Faith, since the pope is the living rule of the Faith that the Church has to follow, and that in fact, She always follows."
He continues:
"God some times can allow that the vacancy of the Apostolic See be for a certain time. He can allow also that a doubt may come concerning the legitimacy of such-and-such an election, but He cannot allow that the whole Church accept as a pontiff one who is not really legitimate. Therefore, from the moment that the pope is accepted by the Church and is united to Her as the head to the body, we can no longer raise the doubt on the possible bias of election or the possible lack of the necessary conditions for legitimacy. Because this adherence of the Church heals in its root all faults committed at the moment of election, and proves infallibly the existence of all the conditions required."
For example, if a Cardinal would have "bought" the papacy (by simony), and the Church accepts the election, that person would be validly pope. There is strong evidence to suggest that this, in fact, did happen with the election of Pope Alexander VI.
St. Alphonsus states that:
"It doesn’t matter that in past centuries some pontiff has been elected in an illegitimate fashion or has taken possession of the pontificate by fraud: it suffices that he has been accepted after as pope by all the Church, for this fact he has become the true pontiff."
St. Alphonsus follows the principle that if the whole Church, and mainly the clergy of Rome accept this man as pope, the man is the pope.
Another authority to which we will refer is Dom Prosper Guéranger, the Abbot of Solemnes, and the great 19th century authority on the Papacy, whose study, Pontifical Monarchy, helped Pius IX to make the definition of Papal Infallibility. In his Liturgical Year, for the feast of Pope St. Silverius, whose election to the pontificate was doubtful, Dom Guéranger writes,
"The inevitable play of human passions, interfering in the election of the Vicar of Christ, may perchance for a while render uncertain the transmission of spiritual power. But when it is proved that the Church, still holding, or once more put in possession of, her liberty, acknowledges in the person of a certain Pope, until then doubtful, the true Sovereign Pontiff, this her very recognition is a proof that, from that moment at least, the occupant of the Apostolic See is as such invested by God himself." (Abbot Guéranger, O.S.B., The Liturgical Year, Vol XII, pg. 188)
Silverius’ pontificate was doubtful because it was forced by the hand of an Arian Emperor. Abbot Guéranger holds that the Roman Clergy would have been free to reject the pope elected as an impostor, since he was thus put in office; but because Silverius was a good and worthy man, and because he was unaware of the violence and evil which brought about his election, they accepted him — and by that acceptance, he was the true Pope.
Therefore, by this principle and the doctrine of the Perpetuity of the Papacy, John XXIII, Paul VI, John-Paul I, and John-Paul II, have been elected to the Chair of Peter, regardless of their supposed illegitimacy. Because they were accepted by the Visible Church as pontiffs, they became true popes.
http://www.catholictradition.org/Tradition/indefectibility.htm#2