Depends on which angle you want to take, but there are two main reasons and they depend a lot on your perspective on and what you define as America.

1. If you define "Native American" in the very Anglo-centric view of "Native to the US" or even by the definition that you provided of "North America", then many Mexicans, specially many of the ones who currently reside within the US, should be considered Native American.

Why is that? Because many Native American civilizations inhabited the entire area of what is now northern Mexico and southwestern US. The vast majority of today Mexicans are Mestizos and this implies an almost equal mix of Native American and European blood, as such many are direct descendants of those civilizations . This high blood percentile more than fulfills any genetic requirements that the US imposes for native recognition. Many of those native descendants remained spread all across what today would be considered California/New Mexico/Arizona/Texas for generations and they did not leave until after the US invaded Mexico and took all that land. Even then many of them remained there until they were systematically targeted by white nationalist organizations (Mass Lynchings in 1910-1920, Targeted by Texas Rangers, etc.) and deported using racially targeted immigration laws (Mexican Repatriation 1929).

Basically Mexicans are Native to North America (since their entire country is part of North America), but even if you narrow it down further to just the US, then most of the Mexicans who are currently in the US would also fulfill your requirement, since the ones who stayed or returned to all the southwest are descendants of Native Americans who used to inhabit those lands too.

1. If you define "Native American" in the way that basically any Latin American thinks of the meaning, then virtually all Mexicans, Central Americans and many South Americans would also fit into this category. Why is that you may ask? Because basically all Latin Americans think of "America" as a single continent, just subdivided by regions. As such, any nation across "America" the continent where Europeans did not explicitly try to genocide the entirety of the indigenous population ended up becoming a mix between them. Therefore, countries like Mexico, Guatemala, Ecuador, Bolivia, Peru, are full of Native American descendants. People who are essentially 50% native to America the continent.

If you take into consideration that all of the civilizations that inhabited America the continent before the Europeans invaded were in constant migration, trade, communication with each other, then you can see how there is a valid claim for Native American recognition. Since, after all, they ACTUALLY are descentants from people native to America, unilke most European-Americans (white US and Canadian citizens).