In pre-Columbian Mesoamerica, the issue of population density remains controversial, and I personally think densities and absolute numbers have tended to be exaggerated in some circles, for some areas. I am not qualified to say much about Mesoamerica outside of the Maya region, but I did recently look into this question for my dissertation. I went into it thinking that populations were pretty high, and came out of it basically convinced that there were not more than three million people in the whole Mayan-language-speaking region (roughly speaking, the Yucatan Peninsula, the lowlands of northern Guatemala and Belize, and the mountainous highlands of Guatemala and northern Honduras) at its pre-Hispanic population peak, around the eighth century A.D. That was just before the infamous "Maya collapse" and well before the Spanish Conquest.

The big problem is that, even though the lowland Classic Maya had a writing system and a well developed historical tradition, they did not record population numbers in any media that have survived to the present day. That means we have to turn to archaeology to address the question - and, to be sure, we'd want to use archaeology even if we had an ancient Maya census document, just to double check. Archaeologists use counts and densities of what are presumed to be residential structures as proxies for counts and densities of people: we use survey techniques (see below) to find as many structures as we can, assume X number of people per household, count the number of household units we think we see in the archaeological record, and voila! there's your population.

Except that Classic Maya domestic structures tend to be arranged in clusters around a central patio, so does one building equal one household, or does one patio group equal one household? And since we have no documents telling us how many people tended to make up a "household" in ancient times, we have to rely on ethnographic accounts of post-contact Maya households in towns not yet transformed by the industrial age - and there, the numbers of people per household range from 3.8 to 25. One widely used estimate is an average of 5.4 people per household. But then you have to factor in whether all the structures you can detect are supposed to have been in use at once; a reasonable estimate is that about 10% were temporarily disused at any given time. And of course not all domestic structures were sleeping spaces, and there, estimates of how many structures in domestic compounds were non-residential range from 6.6% to 30%. Even worse, the Maya lowlands are a terrible place in which to try to find an average commoner's 1300-year-old farmhouse. That house was probably made of wood and wattle-and-daub on top of a low platform of earth and unworked or roughly worked chunks of limestone, and now that there's jungle all over everything, the remains are going to be a bit hard to detect.

These problems have led to some pretty divergent, and in my opinion sometimes highly inflated, population estimates. Let's take as an example the city of Caracol, in modern-day Belize - one of the most politically powerful, and probably populous, Classic Maya cities, although not among the very most populous. Think of it as an ancient Maya Boston, not a New York. The archaeologists who have worked at Caracol for decades have long favored a minimum population for the city of 115,000 people in a 117 sq km area at its height, although they believe it was actually much higher, maybe around 150,000. They arrived at this number in the mid-1990s and have basically stuck with it since then. That estimate means a population density of about 651 people per square km in "downtown" Caracol, which is a little bit denser than Taiwan was in 2012.

Recently, they were able to use some grant money and newly available technology to conduct an aerial laser scan of the ground surface in their site area. The results of the scan are fantastic - 4,732 residential compounds (made up of multiple buildings) were detected in an area of 113 sq km. Many of these were not previously known about, so this technique (known as lidar, like radar with lasers) clearly delivers real results. The thing is, if you (1) assume that all the households were occupied at the same time and (2) multiply those 4,732 compounds by 5 people per household, you get 23,660 people, or a density of a little more than 209 people per sq km in that core area. Since probably not all domestic compounds were occupied at the same time, the actual numbers would likely have been a little lower. These numbers are in line with what the Basin of Mexico project (W. Sanders, J. Parsons, and R. Santley, 1979, The Basin of Mexico: Ecological Processes in the Evolution of a Civilization. Academic Press, New York) proposed as the absolute highest population density for Central Mexico under the Aztec Triple Alliance, just before the Spanish Conquest.

Now assume, optimistically, that the lidar survey only picked up half the households, and there are really more like 9,464 in the survey zone. That still only gets you to 64,355 people, or about 354 people per sq km. Given that the lidar almost certainly didn't miss half the buildings in the survey zone, and given that they surveyed the densest part of the city (so that expanding the survey zone would give you diminishing returns in terms of the number of additional households you'd find), I think it's safe to say that Greater Metropolitan Caracol, including its suburbs, was home to somewhere around 60,000 people at its absolute peak.

Well, so Caracol's a pretty typical very-large-but-not-the-largest Classic Maya city. If we assume similar numbers for similarly sized cities, and correspondingly larger and smaller populations for larger and smaller cities, the maximum total population of the central and southern Maya lowlands at its peak in the eighth century A.D. would have been less than two million. David Webster, an archaeologist just retired from the University of Pennsylvania, favors a figure in the high hundreds of thousands, because he thinks the Classic Maya relied heavily on swidden agriculture; it was actually an essay of his that got me seriously thinking about this issue, and induced me to look at the Caracol data. Other scholars whose opinions on Maya agriculture I also respect believe that agricultural intensifications could have raised those numbers substantially. Still, you're looking at not that much more than a million people when you take all the southern lowland cities into account, and adding the northern lowlands and the highlands probably doesn't even get you to three million. (I think this is so because the northern Yucatan Peninsula is a pretty agriculturally marginal area that was never able to support as many metropoli at the same time as the southern lowlands were, even though some northern lowland cities, like Chichen Itza, were evidently very populous in their day.)

I expect to be challenged on these claims, and I should be, and I'll post sources tomorrow as it's getting pretty late where I am. God knows I have them in spades.

TL;DR: There's no data file you can download.

Very little. /u/cociyo does a good job of illustrating some of the difficulties facing any attempt at ancient demography, but the undercurrent to it is that Classical Maya is one of the best known region in the world for that time. We can probably come up with better figures for China and the Mediterranean region, and perhaps figures of comparable accuracy for northern Europe. Estimates for India, which must have housed a substantial portion of the world's population, do not approach that quality, nor do they for southeast Asia. And even still I am leaving out great swaths of the world's territory, such as Eastern Europe, west Africa, and central Asia.

So consider first that the Maya figure is hotly contested and ranges by an order of magnitude, then that it is comparatively well understood.