I think that our culture might have been influenced by Native American beliefs longer. Even if European expansion was inevitable, and I believe it was, the tribes could possibly have been stronger. If not, we would have been influenced by whatever culture did colonize this continent. If Native American tribes developed greater technology and a more centralized government somehow, they might have prevented it. Without some kind of evolution like that we simply would become like whoever conquered us.
The scope for a alternative, parallel universe novel is immense! I got there first though, and I will sue anyone who steals my idea... only joking...
No, seriously, I do think it is a matter of if not the Europeans then who else. It is not beyond the scope of imagination to consider an Asian Dynasty successfully colonising the New World and having a massive impact in the same way that Europeans did. However, the Europeans, in their role as world leaders at the time, definitely were the ones who were most likely to achieve this expansion.
My understanding is that Canada did not "conquer" the Native Americans, but rather made deals to buy their land. You might take this "bargain" as a starting point and then look at how it has been interpreted and debated.
I am not so sure about the demise of the Native Americans if a more enlightened and culturally sensitive band of settlers arrived instead of the Europeans; perhaps a group who had more compassion and/or understanding of the nation's original peoples. I would use the decline and then resurgence of New Zealand Maori after the pakeha arrived here as an example of how things could have been different in the US.
There was a severe decline in the Maori population due to disease and war when settlers first arrived. Then, however, as the settlers took an interest in assisting Maori to improve hygiene, nutrition and sanitation in their communities the death rate dropped and numbers of Maori increased. There was by no means harmony, and lands were siezed from Maori as from Native Americans, but there exists now a greater parity between Maori and Pakeha than between Native Americans and those of European descent.
It is possible that if the colonists had a different outlook when they arrived in the Americas than the Native American population would be more widespread and prevalent.
We would definitely be more Native American (Eskimos, other tribes having walked over on the landbridge before land masses shifted and cut away from Asia)...perhaps darker eyes, darker hair and skin, more homogenous than heterogenous? The prospects are beyond interesting and intriguing.
I have to agree with the above posts. If the Europeans had not started exploration and settling in the Americas, someone else would have. I think that the fate of the Native Americans would have been similar in either situation.
I often ask my students to write an essay as a Summer Assignment, how would the History of the United States have been different if it had been settled first on the West Coast rather than the East. There is a tendency to believe that European settlement of the Americas was inevitable (several students even postulated that it would have taken longer.) But to address the question; with the expansion of the World as it was and is, the Americas would never have been left a pristine wilderness. There is evidence that Huang He, the great Chinese navigator, landed on the coast of South America. Had it not been the Europeans, then the Americas would most likely have been settled by people of Asian descent. Their plethora of natural resources and wide open spaces meant that exploration, colonization and settlement of the Americas was inevitable. The only question would be "by whom."
A variation on this theme was explored in Kim Stanley Robinson's, "Years of Rice and Salt." The premise is that the Black Plague wiped out nearly the entire population of Europe in the 14th century, leaving the civilizations of Asia room to grow and expand. The dominant world powers that develop are China and the Islamic world, though there are other powers - he puts steamships in the hands of a conquering ruler of Kerala in south India, and a Native American nation rises in North America. If I remember right, though, the Japanese colonize the American west coast. It roughly follows the contours of actual history in a parallel form. I should add, too, that the thread holding it together is a group of characters who reincarnate together periodically, showing up as different people to work together (or sometimes against each other) as history moves forward. Well worth the read for alternate history.
But to respond to the "virgin fields" issue of epidemics, this isn't explored beyond the initial premise of the Black Plague. I suppose there might be some sort of rationalization in seeing East Asians "discover" the New World in this book, although more likely, the author didn't want to deal with it.
I think we would see a world more dominated by Asian economic and military empires than European ones over the three centuries following the rediscovery of the Americas. Whether or not those Asian empires ended up colonizing or exploiting North and South America, European nations that did not benefit from the wealth and resources of the Americas would have been decidedly less competitive in the world arena.
There is, of course, no right answer to this because we can never know for sure...
I am going to assume that Europeans did make contact with the "New World" in your new scenario but did not conquer it. (I am assuming you're talking about the New World here because you have the question tagged with "Cortes.")
If that is the case, there is at least one thing that would have remained the same -- the Native Americans would still have died in huge numbers. One of the major causes of death among the Native Americans were the "virgin field" epidemics of things like smallpox that came over to the Americas as part of the Columbian Exchange. Even if the Europeans had not conquered the Americas, huge numbers of Native Americans would have died.
After that, I think that the remaining native kingdoms would have competed with one another. Whichever ones got enough European technology and modernized quickest would have come to dominate, much as Japan came to dominate Asia because it modernized first. Because the European technology and forms of organization would have been so important in determining who came to dominate the Americas, I think that the world would still be dominated by the technology and ideas of the Europeans the way it is today.
The main thing that would be different is that the United States would not exist. This would be a huge difference because it would mean that the European countries would probably still dominate the world.
There are so many ways this could have played out, though. This is just my take...