I strongly believe that we humans will have to go to space, as I strongly doubt that God made an uncountable number of empty worlds for no reason.
BUT, there are two good arguments that suggest that we won’t be doing so before the Second Coming, due many thousands (or hundreds of thousands…) of years from now.
FIRST, the Dominion Covenant of Genesis 1:26-31 makes no mention of men ruling other worlds:
Then God said, “Let Us make man in Our image, according to Our likeness; let them have dominion over the fish of the sea, over the birds of the air, and over the cattle, over all[b] the earth and over every creeping thing that creeps on the earth.” So God created man in His own image; in the image of God He created him; male and female He created them. Then God blessed them, and God said to them, “Be fruitful and multiply; fill the earth and subdue it; have dominion over the fish of the sea, over the birds of the air, and over every living thing that moves on the earth.”
And God said, “See, I have given you every herb that yields seed which is on the face of all the earth, and every tree whose fruit yields seed; to you it shall be for food. Also, to every beast of the earth, to every bird of the air, and to everything that creeps on the earth, in which there is life, I have given every green herb for food”; and it was so. Then God saw everything that He had made, and indeed it was very good. So the evening and the morning were the sixth day.
An argument can be made that other worlds could be considered other earths. And, with our accelerating technology and with post-millennial assumptions that we have several thousands of generations to go (in a gradually improving, increasingly godly humanity over the centuries), it is likely that we should be able to terraform worlds before the year AD 3000.
But still, the plain sense of the reading can be reasonably read to assume that there is only one earth we can live in: even terraforming Mars – the easiest world – is exceedingly expensive. It would be far cheaper to comfortably house trillions in the sea, under the earth, or in earth-bound towers than to properly terraform Mars, complete with magnetic core (Even assuming radically inproved 3D printing, nanotech, and the rest of the ‘far-future’ technologies.)
SECOND, wherever men go, so does war.
Right now, there is a sharp decline in top-tier warfare, as nuclear weapons make such a war very fatal for the power-elites that typically benefit from such conflicts. The failure of war has spread, to the extent that the U.S. military, the most powerful armed force on the planet, cannot decisively win a conflict against ~4,000 or so guerrillas in Afghanistan.
(Elsewhere, it is more potent – see the recent French war in Mali – but even the French have no interest in staying long-term in that nation. Meanwhile, in Syria…)
The ability to travel to other worlds imply the ability to sterilize said worlds, by nuclear-tipped weapons, warp-drive weapons, factional-c weapons, grey goo weapons, and whatever cruel devises sinful men can come up with.
This suggests that, soon after any interstellar empire is built, said empire will be replaced with a large number of radioactive ruins and slaughtered populations. It is reasonable to assume that such slaughter will leave humanity far worse than if they had simply remained on Earth, fearful only of the murderous tools we have already created by our own hands, instead of inventing new ones.
I don’t believe that men can do that.
Just because humans are innately evil, though, does not mean that we can’t expand into space to some extent before joyfully slaughtering ourselves over some laughable justification or other. If FTL travel remains impossible, the cost of interstellar war soars beyond conception, making it impossible for all for the foreseeable future. And even our evil nature is increasingly restricted, with steadily declining violence over the last millennia. No one ever managed to have a large-scale revival of the Roman gladiators, the Aztecs are gone for good, and it’s unlikely that either forms of Socialism will ever again have a mass following.
It’s unlikely that widespread abortion will outlast the destruction of our materialistic Lord and Saviour, the State, even assuming the universal ability to 3D print as many abortion pills as you please. The children of the future will be descended from religious parents with a rigid moral code, not sterile secularists who explicitly deny any unchanging law beyond the authority of the State.