The Fathers have opined on the question of whether God would allow the entire human race to become celibate.

St. Gregory of Nyssa (370 A.D.)

St. Gregory of Nyssa thought that if all of mankind embraced continence, then God would create new humans in a manner similar to how he created Adam, out of the earth. St. Thomas Aquinas summarizes St. Gregory's view in Summa Theologica I q. 98 a. 2 ("Whether in the state of innocence there would have been generation by coition?") co.:

Gregory of Nyssa says (De Hom. Opif. xvii) that in paradise the human race would have been multiplied by some other means, as the angels were multiplied without coition by the operation of the Divine Power. He adds that God made man male and female before sin, because He foreknew the mode of generation which would take place after sin, which He foresaw.

St. Thomas disagrees:

But this is unreasonable. For what is natural to man [e.g., sexual intercourse] was neither acquired nor forfeited by sin.

St. Ambrose (378 A.D.)

To the objection that St. Ambrose was depopulating the Roman Empire by his zealous advocacy of virginity and the religious life, he replied implying there will always be marrieds (On Holy Virginity ch. 7):

Finally, is it [virginity] inexpedient? The world is alarmed for itself, as though marriages are to be no more and the human race to be extinct. But what need of terror; has there ever in fact been a lack of persons willing to be brides? And as to ills arising from either state [virginity or marriage], broken wedlock has been the source of wars and destruction, but to none has consecrated virginity occasioned death, for it is higher than human law and beyond its sanctions; religion gives it its dignity, and its safeguard is the Faith.

and that the population actually grows where virginity is more highly esteemed (ibid.):

Nor is there fear of the extinction of mankind. Facts have proved the contrary, and shewn that where virginity has been most honoured, there most mankind has multiplied. What multitudes of virgins were yearly consecrated at Alexandria, in the African Church, and throughout the East [in the time of St. Ambrose]. Births were rarer in the West, than consecrations of virgins in the East. Even on this ground, then, let not virginity be thought unprofitable; yea, how profitable has it been, seeing that by a Virgin salvation came, to render fruitful the Roman world.

If ye will urge this futile argument, pause and see whither it will carry you. It will lead you to defend the violation of wedlock, if thereby mankind may be more multiplied.

The wedded need not be alarmed, they have their wives; for the unwedded man, whom has he to blame but himself, if he seek a maiden who he knows beforehand will not be a bride? Fathers need not fear, for if other maidens are consecrated, their daughters may be the sooner chosen.

St. John Chrysostom (381 A.D.)

In On Virginity ch. 15 ("Marriage does not increase the human race."), St. John argues that sin, not virginity, causes death:

1. And today our race is not increased by the authority of marriage but by the word of our Lord, who said at the beginning: "Be fertile and multiply; fill the earth." [Gen. 1:28] How did marriage help Abraham in the procreation of children? After participating in it for so many years, did he not finally cry out: "Master, what will you give me? Am I to die childless?" [Gen. 15:2] […] For marriage will not be able to produce many men if God is unwilling, nor will virginity destroy their number if he wishes there to be many of them.

St. Jerome (406 A.D.)

St. Jerome wrote against the heretic Vigilantius, who opposed virginity:

to follow out your [Vigilantius's] argument, virginity would not deserve our approbation. For if all were virgins, we should have no marriages; the race would perish […] The truth is, virtue is a rare thing and not eagerly sought after by the many. Would that all were as the few of whom it is said: “Many are called, few are chosen.” [Mt. 10:16, 22:14] The prison would be empty.

St. Thomas Aquinas, quoting what St. Jerome wrote here, agrees that virgins and others (e.g., the religious, bishops) living in a state of perfection will be rare (Contra impugnantes Dei cultum et religionem, An Apology for the Religious Orders II cap. 6 ad 12):

The works of perfection are so difficult that but very few attempt to accomplish them. There is, therefore, no grounds for fearing that the world will cease to exist on account of the perfection of its inhabitants.

Would God allow all of humanity to be celibate?

The short answer is no.

This goes directly against what God said to Adam:

28 And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth. - Genesis 1:28

God would not allow humanity to disappear by such an act to transpire.

Besides man has free will and as such it would be extremely doubtful that every man, woman and child would voluntarily remain celibate for life.

The Catechism of the Roman Catholic Church asserts that "Freedom is the power, rooted in reason and will". It goes on to say that "God created man a rational being, conferring on him the dignity of a person who can initiate and control his own actions. God willed that man should be 'left in the hand of his own counsel,' so that he might of his own accord seek his Creator and freely attain his full and blessed perfection by cleaving to him."" The section concludes with the role that grace plays, "By the working of grace the Holy Spirit educates us in spiritual freedom in order to make us free collaborators in his work in the Church and in the world." - Free will in theology