In a Catholic viewpoint, is it possible for Rome to fall? I was wondering if the Roman Catholic Church is subject to complete institutional corruption, or total destruction, according to Catholic thought.

Now this would not necessarily be the fall of the Catholic faith, but instead that of its earthly headquarters and global authority.
I want to know because it may play a role in personal view of the end times, and the view of Catholics as well of course.
From the Catechism of the Catholic Church:


  754 "The Church is, accordingly, a sheepfold, the sole and necessary gateway to which is Christ. It is also the flock of which God himself foretold that he would be the shepherd, and whose sheep, even though governed by human shepherds, are unfailingly nourished and led by Christ himself, the Good Shepherd and Prince of Shepherds, who gave his life for his sheep.Lumen Gentium 6; Cf. Jn 10:1-10; Isa 40:11; Ezek 34:11-31; Jn 10:11; 1 Pet 5:4; Jn 10:11-16.


The Church is a community established by the Lord Himself, and He unfailingly nourishes and Leads her; if she were subject to "complete institutional corruption," as you put it, then it would mean that the Christ failed to nourish and lead her.


  765 The Lord Jesus endowed his community with a structure that will remain until the Kingdom is fully achieved. Before all else there is the choice of the Twelve with Peter as their head.Cf. Mk 3:14-15 Representing the twelve tribes of Israel, they are the foundation stones of the new Jerusalem.Cf. Mt 19:28; Lk 22:30; Rev 21:12-14 The Twelve and the other disciples share in Christ's mission and his power, but also in his lot. Cf. Mk 6:7; Lk 10:1-2; Mt 10:25; Jn 15:20 By all his actions, Christ prepares and builds his Church.


The structure of the Church was endowed by Christ and will remain until the completion of the Kingdom, and therefore isn't subject to complete corruption unless the Kingdom has already been fully achieved.


  768 So that she can fulfill her mission, the Holy Spirit "bestows upon [the Church] varied hierarchic and charismatic gifts, and in this way directs her."LG 4 "Henceforward the Church, endowed with the gifts of her founder and faithfully observing his precepts of charity, humility and self-denial, receives the mission of proclaiming and establishing among all peoples the Kingdom of Christ and of God, and she is on earth the seed and the beginning of that kingdom."LG 5


The Holy Spirit is responsible for directly guarding and protecting the Church by dispensing His gifts, both hierarchical and sacramental. It is the Holy Spirit who guarantees both Conciliar and Papal Infallibility, by preventing them from teaching in their Canons or ex Cathedra, respectively, anything which will corrupt the Church.

In any case, there is also the Biblical promise in Mt 16:18:


  18 And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.


Here Christ clearly says that the Church shall subsist until the end of time, so we are justified in believing that she will not fail.

Now, regarding the city of Rome itself, or the temporal power of the Papacy, that might (and has, at times) fallen, but the point of the Vatican existing is so that the Church isn't seen as an Italian institution, and the Pope isn't seen as a subject of the Italian State. For contrast, recall that the Ottomans would not allow a foreigner to occupy the Patriarchate of Constantinople after they conquered that great city.

In any case, the fall of the Vatican City State does not imply the fall of the Pope's global authority, though it might affect his standing among the diplomatic corps of the world. Even if he were chased out of the Eternal City and had to take up residence in Pisa, Avignon, Dharamsala or New Jersey, he would remain the Supreme Pontiff, Vicar of Christ, and the Rock upon which the Church is built.