The Catholic answer, as best I can.
Firstly, if your question means to ask, can the current lineage of humanity transition into a state of complete grace, free from the burden of original sin or the fallen nature of the world, the answer is a simple no. The doctrine of original sin precludes it: it's a fallen world. And God has already revealed a different solution.
However, I think that "different solution" is worth digging into a little. Because your question could mean, could God's Kingdom arrive in the universe as we know it, transforming everything and everyone at once, or does it need to occur in a completely new universe?
So in brief, there are two relevant Catholic beliefs that I think provide an answer to the 2nd question, if that's the question you meant.
Firstly, as stated in the Catechism of the Catholic Church (CCC), we believe in the renewal of all things. This belief, if I'm not mistaken, is taken primarily from The Revelation of John, a few of Christ's own words, and from some OT apocryphal segments.
At the end of time, the Kingdom of God will come in its fullness.
After the universal judgment, the righteous will reign for ever with
Christ, glorified in body and soul. The universe itself will be
renewed:
The Church... will receive her perfection only in the glory of heaven,
when will come the time of the renewal of all things. At that time,
together with the human race, the universe itself, which is so closely
related to man and which attains its destiny through him, will be
perfectly re-established in Christ.
The renewal is later said in the Catechism to be a sort of transformation.
The visible universe, then, is itself destined to be transformed, “so
that the world itself, restored to its original state, facing no
further obstacles, should be at the service of the just,” sharing
their glorification in the risen Jesus Christ.
Secondly, we believe all people are born into a state of original sin, with the exceptions of Jesus and His Mother, Mary. The basic concept of orignal is summed up nicely in the CCC:
Although it is proper to each individual, original sin does not
have the character of a personal fault in any of Adam’s descendants.
It is a deprivation of original holiness and justice, but human nature
has not been totally corrupted: it is wounded in the natural powers
proper to it; subject to ignorance, suffering, and the dominion of
death; and inclined to sin—an inclination to evil that is called
“concupiscence.” Baptism, by imparting the life of Christ’s grace,
erases original sin and turns a man back toward God, but the
consequences for nature, weakened and inclined to evil, persist in man
and summon him to spiritual battle.
It goes on to imply that Adam's sin has an impact on all mankind, and therefore the world of which he is the steward, because he handed his freedom, to some extent, to the devil, who only then is given real power in the world.
Put these doctrines together and a picture begins to form. Mankind relinquished his power in an appeal to Satan. Satan is thereby given reign in the world, since it's under Adam's (man's) stewardship. And because this affects the universe on a fundamental level, it becomes the nature of the universe. Thus, the only possible solutions are a completely new universe or fundamental transformation of the universe.
Now, since the Catholic teaching seems to lean towards a transformation or a renewal, we can draw some conclusions in light of what we also know about the fallen nature of the world.
- Everyone must die. Necessary because no one is born free of original sin, and our bodies and minds are corrupt. So, we need to give them up.
- Everyone must then be raised. (Or at least the elect.) If they aren't, we haven't got an eschatology at all! Just a bunch of dead people and a lingering fallen world.
- Our new bodies must be made perfect. Refraining from the more speculative ideas as to what that really means, our bodies and souls are in perfect harmony with each other and with God.
- The universe must be transformed into perfection. I don't know what this means! The Church, I think, suggests that it frees the universe from the "decay."
- Satan must be confined to hell. He no longer gets to operate in the world of the elect.
So, it's not beyond the realm of possibility that the fundamental nature of the world changes without changing its immediate appearance. But, it is necessary that we all die. And it is necessary that world-processes be freed from Satan's reign, which likely means that things work somewhat differently.
All things considered, I think the answer to your question is still no.
For, even if the universe is renewed in-place, on-the-fly, we're still dealing with a fundamentally different universe. And we've all had to die, be resurrected with perfected (different) bodies, etc.. We're dealing with something that, in our limited understanding is already very different, even if in God's eyes it's the same material.
