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If you check a list of the most densely populated nations, a lot of New World countries are near the bottom and almost no Western Hemisphere nation (excluding small islands) are near the top. The Americas was depopulated during the Columbian Exchange and even the Pre-Columbian Americas were sparsely populated in a lot of regions. As a result, the regions of the New World aren't densely populated like many regions of the Old World. There hasn't been enough time and immigration since the colonial era to make many countries there densely populated. And considering declining birth rates throughout the world, North America or South America will probably never be as densely populated as Europe and Asia is.

Let's say though that in another universe, the Native Americans were far more advanced. Even though they had less time to develop their societies and were isolated, by some miracle, the pre-Columbian polities end up achieving technological and sociological parity with Eurasia. By 6th Century AD, the Native Americans have the same technological base and similar political institutions to Europe. The Native Americans also have the same domesticated animals East Asia does due to different fauna migrations. As a result, the Native Americans don't suffer diseases from the Columbian Exchange and they also don't get colonized.

Given these factors what would be the estimated population of North and South America by the 21st Century? North and South America would probably have a far larger population than Europe thanks to their increased size. That said, you can't just do a simple population density calculation because there are many areas of the Americas like the Alaska and the Amazon Rainforest that would be hard to live on. The Native Americans also have access to different crops than the Eurasians do. You do not have to factor in epidemics or wars or natural disasters for your population estimate.

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    $\begingroup$ If the Americas had the same level of civilization as Europe there would be no Amazon Rainforest. In the 6th century, Europe north of the Alps was a continuous almost uninterrupted forest from the Atlantic in the west to the Dnieper in the east. The forestless agricultural Europe we all love was made by humans. (East of the Dnieper there was no forest because that part of Europe is steppe, like the American Great Plains.) $\endgroup$
    – AlexP
    Jan 7 at 22:59
  • $\begingroup$ Possibly the same, or much more or much less, depending on how the conflicts went post 1492. But you want it given that "they also don't get colonized." Big given... $\endgroup$
    – user458
    Jan 8 at 2:53
  • $\begingroup$ Is it unclear if you want us to assume that the disease apocalypse and depopulation did not occur. $\endgroup$
    – user535733
    Jan 8 at 16:38
  • $\begingroup$ Is it plausible that they'd have the same illnesses as Europeans, and thus get immunity? I imagine instead each continent would have a different set of really bad illnesses, exchange them, and we'd get two apocalypses (apocalii?) $\endgroup$
    – Zwuwdz
    Jan 8 at 19:16
  • $\begingroup$ You might want to read Guns, Germs, and Steel. Diamond (the author) posits that technology is only a small part of the story, and that the real story is much more complicated. While technology had its place (European metallurgy, more than guns, which were primitive and inaccurate at the time), horses, disease, and geography were much more important. $\endgroup$
    – Xander Henderson
    Jan 9 at 13:51

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even the Pre-Columbian Americas were sparsely populated in a lot of regions

It was mostly in hunter-gatherer phase

The Native Americans also have the same domesticated animals East Asia does due to different fauna migrations.

You don't need that. It's enough that they managed to domesticate some of their existing megafauna, before driving it in to extinction. In RL among others they actually had horse.

"Given these factors what would be the estimated population of North and South America by the 21st Century?"

Deciding factor - potential to produce enough food to feed this population in pre-modern times. (Diseases are overrated as long term factor. Sure, you get pestilence from time to time, half population dead, followed by population boom as next generation is suddenly well nourished)

A few billion, like in Asia:

  • surface is comparable, conditions are comparably varied
  • in pre-modern times the were most lucky concerning staple crops - potato & corn, which had potential for population explosion as Asian had with rice.
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Technology isn't really the deciding factor here.

To begin Population growth takes time, regardless of available technology. It is unlikely they would grow any faster, particularly as the new world per-colonization had less of a problem with disease.

On that note, It's worth noting that there actually were many natives when they were first encountered, but a signficant percentage of them were wiped out by European diseases. In fact, the Americas that the Europeans discovered could be described as post-apocalyptic, as their diseases had colonized the new world much quicker than they themselves could.

The reason Europe didn't suffer a similar fate was largely due to the various animal species of the two continents; Europe had more domestic animals, which resulted in more disease; The native Americans did not have so many viable candidates for domestication, which meant they had less disease. This CPG Grey video goes into further details.

https://www.youtube.com/watch?v=JEYh5WACqEk

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    $\begingroup$ That is all true but doesn't given an answer to OPs question. $\endgroup$
    – quarague
    Jan 8 at 7:14
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At the time of the conquest, some cities in the New World were huge. Mexico City was estimated to have 100,000 to 200,000 inhabitants. That is larger than the cities in Europe at that time. The population of the Mississippi basin was quite large. There was a very significant civilization in the Amazon basin, an area that is now forest. One can make the claim that the Native Americans were more advanced than the conquering Europeans.

One answer would be to take the high estimates of the population and use that.

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Short answer: Maye a bit more than today but not that much.

At a modern technology level, populations can increase by 5%/year and more (Link, note that most countries in this chart are actually struggling with medicine and food supply). If this growth rates would have been reached for only a few hundred years, we would look at a global population of trillions. So technology is the deciding factor. Only when technology reduced the deaths from famine and diseases substantially, population growth really kicked in. This happened almost exclusively during the late 19th and the 20th century (Link). This time is the reason why the global population is as high as it is today. The time before it hardly matters, because the global population was almost stagnant before this time (compared to todays numbers) due to diseases and famines.

In this time, the Americas (especially North America) actually had the same technology level than Europe, so your scenario is not that different from real history regarding population size.

Some areas in North America (northwestern USA and Canada) were very scarcely populated in the mentioned timeframe, so population in your scenario might be a bit higher. Also, in South America the majority might have better access to food/medicine in your scenario than it had in history (were food/medicine was probably mostly restricted to rich people), so population size might also be a bit higher.

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