You don't think that people from Europe would have emigrated to the US?
The historical chain of events that led to the Age of Discovery for the Western European powers wouldn't exist.
No Christianity means that the Roman Empire never adopted Christianity, it means that Christianity never spread among the Celtic and Germanic tribes of Europe. The Visigoths, Franks, Lombards, Anglo-Saxons etc never became Christian.
It's entirely unclear what Late Antiquity would look like without Christianity, what would have happened to the Roman Empire? Would Constantine still successfully unified the Roman Empire under his rule? Without Constantine the capital probably wouldn't have been moved to Byzantium. Changing the base of imperial power from Rome to Byzantium also had the consequence of, with growing strength of the immigrating Germanic tribes, weakening the Roman Empire in the West and opening the door for the sacking of Rome and the fall of the Western Roman Empire. It's entirely possible that this would have still happened anyway just in a different way.
Further, without Christianity it is unlikely that Islam would exist, without Islam there is no Islamic world. That means that the Eastern Roman Empire didn't lose territory to the Muslim conquests of the early Muslim Era. It also means that the Sassanid Empire never fell to the Muslims either. It means no Ottoman Empire and no Ottoman Empire means no fall of Constantinople in 1453. Without the Fall of Constantinople and the closure of the Silk Road and spice trade with the East, the European powers would have had no needs to start finding new sea routes to the East, such as the Portuguese sailing around Africa to reach the Indian ocean. It also means no Christopher Columbus sailing west to try and reach Asia.
The entire Columbian Exchange wouldn't have happened. Up to that point the only Europeans who had seen the Americas were the Vikings, and their settlements did not last.
That means there is no Age of Discovery. No era of European colonialism. There are no political powers seeking new trade routes for spices with the East because the historic trade routes to the East are still open through the Middle East to Europe. It means no discovery of gold in the New World leading to the conquests and further exploration of the Americas which wiped out the New World powers such as the Aztec Empire.
We can imagine that, eventually, contact will be made. But likely under very different circumstances, and probably much later than it did in our history. There would need to be an impetus, a reason, for some political powers to go exploring. Perhaps a ship caught off course might reach the coast of Brazil by some political power in Western Europe (but this would not be any of the European political powers of our history, since they wouldn't exist). A North African power could also feasibly get off course and reach Brazil too.
What that contact would look like, to what extent the Old World nations would be aware of this being a "New World" like happened in ours is unknown. Would such contact result in trade rather than colonies? Would it result in conquest? It's impossible to know.
But the simple and short answer is that, no, Europeans wouldn't have come to the Americas and establish permanent colonies etc without the historical factors that led to that happening. Search for new spice trade routes, discovery of gold and the search for wealth in the Americas, continued searching for a "Northwest Passage". Without the Protestant Reformation and the splintering of Western Christianity among Roman Catholic and Protestant states and without the pressures of religious persecution or lack of full religious freedom for dissenting religious groups means no settling of the Americas by religious refugees and dissidents, such as Quakers, Puritans, Baptists, etc.
Historical events don't happen in a vacuum. Everything in history is connected with everything else before and after. Changing even a small detail in the past can result in potential "butterfly effects". The butterfly effect, for those unaware, simply refers to the fact that very tiny things can have gigantic unforeseen consequences. Changing obviously major historical events has arguably even bigger butterfly effects.
For example what if Julius Caesar or Alexander the Great had never been born? What if Adolf Hitler died as a baby? What if Temujin never became Genghis Khan?
The further back we go in history, the more divergent history becomes if we change something. Anything, big or small. Even the most seemingly insignificant events have gigantic consequences for the future.
-CryptoLutheran