Was the near-extinction of Native Americans a genocide? If so, why is the US denying it? If not, why not?

I really want to stress that these questions are incredibly loaded and very difficult to answer fairly.

My first and foremost concern is that we should be tremendously uncomfortable generalizing the colonial period with a single term. As /u/anthropology_nerd said earlier:

500 years of interaction with half a dozen European nations and hundred of Native American nations can't be boiled down to an easy answer

However, the US does not deny that various acts of violence occurred. It has issued dozens of apologies regarding specific incidents for decades, which recently culminated in the Apology act. Whether the sum of these incidents amount to genocide and if that term can even be applied to past events is not at all clear. Nevertheless, the US does not really deny the most important and violent elements of its past and present.

However, that doesn't quite add up to the fact that all modern countries in the Americas except for the US and Canada are heavily dominated by people of Native descent, starting right from the Mexican border.

The US-Mexican border is quite unique in the world and I could honestly write for hours on it. One of the interesting things about it is that the majority of the border is not populated by people who identify with officially recognized native groups. The border region was the site of some of the most violent native resistance and subsequent deportation. One example of this is the modern state of Arizona. Of all the counties in Arizona, only 3 lack any Native American reservations. Of those 3, 2 are along the Mexican border and the remaining 1 would be a border county if county lines were based on sanity. The situation was even worse in New Mexico and Texas. All of that said, identity is difficult and I am dramatically simplifying a stupendously complicated topic.

all modern countries in the Americas except for the US and Canada are heavily dominated by people of Native descent

That's not actually true. There are many countries where Amerindians and Mestizos make up a very low percentage of the total population - places like Brazil, Uruguay and most of the Caribbean. The reason why the indigenous population of places like Central America and the Andes is relatively high is because they originally had much higher population densities than most of North America, and they had climates which were not as hospitable for settlement by Europeans as places like Argentina or the US.

I wasn't actually aware that the US denied that the Native American Genocide happened (people like Dinesh D'Souza aside). I'll leave that question for a more qualified person. But your premise of the US and Canada being the only countries in the Americas with a relatively small indigenous population is far from true.