My major has been Russian History, but my minor was Native American / Colonial American History. So I'll try to answer this.
Few myths to get rid of first. I see that someone here has already gotten rid of a few=). Yes, it's very true that the mayan were not predicting the end of the world. The prediction came from people who found a few calender that went to 2012, and assumed because there are so few artifacts of the mayan left, that this was their prediction.
Another myth: "The natives thought Europeons were gods." This has mild truth...Natives only believed this when Europeans were first coming. You need to remember that Natives had never seen the massive warships nor the animals on board. So when they saw giant ships coming towards land, their natural reaction was "Our gods most be coming." This is not necessarily foolish, considerring what Europeans believed in themselves:
The final part of the god myth comes from European cultural egocentrism/misunderstandings. For example, when trading North American natives would call the Europeans "Father." The Europe understood this as "You're saying I am the father, and you're the child.. I must take care of you." When in reality, for Natives... The father was simply the decision maker in trading. This also occurred when Europeans gave Natives Farming tools and beads. Instead of using them, The natives would often wear them / hang them. The Europeans took this as "they're weak and dumb." When for the natives, it was simply a cultural tool for them.
So... Let me try and answer your question. It's incredibly difficult to do.
First, someone asked or stated that Natives would have colonialzed somewhere else at some point. This is false. Historians are pretty aware this wouldn't happen. It simply goes against Native ideology. They do not value land in that way, they were not imperial. And besides their technology for ships could not have brought them to Europe or Africa.
It's also important to note, that early encounters where the Spain "conquered the indians" also has a myth. When the spanish first showed up and attempted to take the Aztecs... The Aztecs kicked their asses. (Far more numbers + Bow and arrows are better than early muskets.) They were only able to conquer them because as they left for 5-20 years to regather forces, 90% of the population died because of small pox.
So the only thing I'd be certain of... had Europeans not contacted the natives.
1) There would be a lot more natives.
2) Europe would be a much more different place (No fuel for mercantilism / empire)
At some point, the entire situation would happen though. North America was virgin soil, none o the diseases had penetrated there. So had Europeans come in say... 1900. 90% of the native population would still collapse.