Below is a list of several prophecies and quotes from the Catechism which definitively state that the "period of peace" promised by Our Lady of Fatima will be extremely short in nature, and should certainly not be considered to encompass an entire "era" during which the Church will triumph for an extended amount of time within history.
If my requests are heeded, Russia will be converted, and there will be peace; if not, she will spread her errors throughout the world, causing wars and persecutions of the Church. The good will be martyred; the Holy Father will have much to suffer; various nations will be annihilated. In the end, my Immaculate Heart will triumph. The Holy Father will consecrate Russia to me, and she shall be converted, and a period of peace will be granted to the world”.
(Second Part of the Secret of Fatima)
France has corrupted the universe, one day it will be punished. The faith will die out in France: three quarters of France will not practice religion anymore, or almost no more, the other part will practice it without really practicing it. Then, after [that], nations will convert, the faith will be rekindled everywhere. A great country, now Protestant, in the north of Europe, will be converted; by the support of this country all the other nations of the world
will be converted.
Before all that arrives, great disorders will arrive, in the Church, and everywhere. Then,
after [that], our Holy Father the Pope will be persecuted. His successor will be a pontiff that nobody expects.
Then, after [that], a great peace will come, but it will not last a long time. A monster will
come to disturb it.
(Our Lady of La Salette to Maximin Giraud)
Now concerning the times and the seasons, brothers, you have no need to have anything written to you. For you yourselves are fully aware that the day of the Lord will come like a thief in the night. While people are saying, “There is peace and security,” then sudden destruction will come upon them as labor pains come upon a pregnant woman, and they will not escape. (1Thes 5:1-3)
After all these calamities have arrived, many will recognize the hand of God on them. They will convert, and do penance for their sins.
A great king will go up on the throne, and will reign many years. Religion will re-flourish
and spread all over the world, and there will be a great abundance. The world, glad to be lacking nothing, will fall again into disorder, will give up God, and will return to its criminal passions.
[Among] God's ministers, and the Spouses of Jesus-Christ, there will be some who will go astray, and that will be the most terrible.
(Our Lady of La Salette to Melanie Calvat)
The brevity of the period of peace is also attested to by the fact that the Church will only be restored during the coming of the Two Witnesses, who are put to death by the Antichrist at
the end of their ministry. Once again, this means that the duration of the period of peace
must be extremely short.
“Behold, I will send you Elijah the prophet before the great and awesome day of the LORD comes. And he will turn the hearts of fathers to their children and the hearts of children to their fathers, lest I come and strike the land with a decree of utter destruction.”
And when they have finished their testimony, the beast that rises from the bottomless pit will make war on them and conquer them and kill them, and their dead bodies will lie in the street of the great city that symbolically is called Sodom and Egypt, where their Lord was crucified.
The upshot of this is to illustrate the fact that according to Sacred Scripture and Tradition, as well as Church approved private revelations, the period of peace is extremely short in nature, and this can in no way be equated with the Millennium of Rev 20. Any Catholics who forward this position are in serious error, since it directly contradicts the teachings of the Catechism, which clearly states that the Church cannot triumph on earth through a
progressive ascendancy during an intra-historical era, and will only enter into the fullness of the kingdom through the final trial of the Antichrist.
The Church will enter the glory of the kingdom only through this final Passover, when she
will follow her Lord in his death and Resurrection. The kingdom will be fulfilled, then, not by
a historic triumph of the Church through a progressive ascendancy, but only by God's victory over the final unleashing of evil, which will cause his Bride to come down from heaven. God's triumph over the revolt of evil will take the form of the Last Judgement after the final cosmic upheaval of this passing world. (CCC 677)
As Cardinal Ratzinger outlined in his book Eschatology:
The rejection of Chiliasm meant that the Church repudiated the idea of a definitive intra-historical fulfilment, an inner, intrinsic perfectibility of history. The Christian hope knows no idea of an inner fulfilment of history. On the contrary, it affirms the impossibility of an inner fulfilment of the world. This is, indeed, the comment contend shared by the various fragmentary pictures of the end of the world offered us by Scripture. The biblical representation of the End rejects the expectation of a definitive state of salvation within history. This position is also rationally correct, since the idea of a definitive intra-historical fulfilment fails to take into account the permanent openness of history and of human freedom, for which failure is always a possibility. In the last resort, such neo-chiliasm expresses a profound anthropological perversion...
Faith in Christ's return is, therefore, in the first place, the rejection of an intra-historical perfectibility of the world.
(Cardinal Joseph Ratzinger, Eschatology, p213)