[kragen]

on the contrary: to the extent that they knew the word, native americans considered themselves americans, and the colonists did not consider themselves americans, but rather spanish, portuguese, english, or dutch, as the case may be

[MavisBacon]

"America" as it was, was simply an idea at that point that europeans were buying in on. And it was not the idea of the natives. Natives identifying as American or something close came much further down the line than the period mentioned in OP. The American Indian Movement is where this all finally crystalized.

[kragen]

"america" was the name the europeans gave to the continent (or pair of continents) discovered by columbus, which he thought was indonesia, but which amerigo vespucci proposed was a new continent. vespucci proposed this after exploring parts of america, specifically parts of venezuela, what is now the dominican republic, and some of the coast of brazil. the name "america" gained currency in europe over the period 01507–01538, 89 years before the tombstone in question and 430 years before the founding of the american indian movement in 01968

the idea that native americans were native to that continent (or pair of continents) was not in any way novel to them; what was novel was that there were other continents such as europe, and thus that there was a distinction between america and the world

so everything you have said is completely unrelated to the truth