This seems like somewhat faulty logic to me. I’ll preface this by saying we don’t know the population of the pre-Columbian Americas with any real degree of certainty - estimates range from ~10 million (almost definitely too low) and >100 million (almost definitely too high). The real number is probably somewhere in the middle, but where exactly it falls really isn’t clear. The consensus seems to be somewhere in the range 40-60 million, with some genetic evidence from the Caribbean supporting lower figures while archaeological and historical evidence from Mesoamerica support higher estimates.

In any case, the one thing that’s not in question is that the Americas have a higher population today than in 1500. But I’d hesitate to link this to colonization. Because, I should stress, this is true of every continent on earth. The European population in 1500 was somewhere in the range of 90 million; today it’s ~750 million. China had a 1500 population of around 60 million; today it’s over a billion. Unlike the Americas, these were by and large* not places where indigenous displacement and genocide opened vast swaths of land to colonial settlement. Instead, it’s much easier to point to things like industrialization as the cause of population growth. Similarly, the world population didn’t reach over one billion until around the turn of the 19th century. It’s now nearly eight billion, with the vast majority of the increase having taken place in the 20th century by which point the majority of indigenous lands worldwide had already been seized and settled in some form.

But let’s look back at the Americas. Specifically, I want to point to instances that demonstrate that population growth does not automatically follow colonization. So, let’s look at Florida, today the 3rd-most-populous state in the US with over 21 million people. Florida was first permanently settled by Europeans in 1565 at St. Augustine, though earlier French and Spanish attempts had occurred (in fact, St. Augustine was an attempt to chase off French settlers near modern-day Jacksonville). At this point, Florida was home to a diverse array of indigenous cultures, from the agricultural, moundbuilding Apalachee in the Panhandle, the semi-agricultural Timucua scattered across its center and southeast, and the fisher-hunter-gatherers with a complex kingdom, the Calusa, dominating the south, in addition to other smaller groups scattered throughout the state. We don’t know what Florida’s population was in 1500 and there’s been a lot of debate, but over the next few centuries disease and forced labor in the missions devastated the Apalachee and Timucua (the Calusa never let the Spanish in). Then, English-Muscogee slave raids from the north killed or captured the remaining indigenous population, turning Florida outside of St. Augustine into a veritable wasteland. Migrants from the north - the ancestors of the Seminole - would fill in the space, but Florida is one of the few places where we can confidently say there were fewer inhabitants in 1800 than 1500. The 1830 census recorded 34,730 residents, admittedly not including the approximately 6,000 Seminole, but the best estimates for the Apalachee alone in 1500 are 30,000 if not higher, and the Timucua were more numerous than them. American interest in Florida led to most of the Seminole being forced west on the Trail of Tears - though some remained in southern Florida, by the mid-1840s most of Florida was “open” to white settlement.

At this point, you would expect the Floridian population to grow. And it did. Just… gradually. The state didn’t surpass one million people until 1930, nearly a century after the close of the Seminole Wars, and as late as 1950 Great Plains states like Iowa had a higher population than the Sunshine State. So what brought people to Florida? A lot of things, of course, but a huge contributor was the invention of air conditioning. It’s how you transform humid swamplands into sprawling metropolises.

Of course, available land plays a role. There’s only so many people you can fit in one area. If you were asking about whether the idea that more available land is likely to lead to a population increase is plausible I’d say it definitely is (though I can’t say whether or not it’s true). But land has to be desirable. People need a reason to live there. Land itself is clearly not the reason for population growth, or Russia and Canada would be far more populous than they are. (For that matter, Russia is a good example of how timing can matter more than availability; Siberia was mapped and settled in the 1600s and 1700s, but it took into the late 1800s and throughout the 20th century to truly populate it like it is today).

So to answer your question: I would not say the colonization of the Americas led to a population increase in the sense that said population increase was not a direct result of said colonization. It’s much more complicated than that.

*You can arguably see this in Sápmi and Ireland in Europe, and in the Dzungar Genocide in Xinjiang in China, but these are small portions of each region as opposed to the entire Americas.