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Elaborating on your point, you should read Guns, Germs and Steel, if you haven't already. The book won the Putilzer prize. In the right climates, Euros really did wipe out the natives. When they tried to colonize the wrong climates, it simply didn't work out.
In regards to Cuba, I do not believe that there was any conscious effort by our early Cuban ancestors to wipe out every last Indian. However, Germs and Geography doomed the Indians in Cuba.
The de facto biological warfare between continents during the Middle Ages and the Age of Exploration was a natural consequence of maritime trade and exploration. European diseases devastated New World indigenous populations. The plague from Asia and syphillis from the Americas ravaged Europe. With some exceptions, such as giving Indians smallpox tainted blankets as was done in some North American Indian conflicts, there was nothing sinister about it. Germs were simply doing their job and the humans had very little say in the matter.
In a large land mass such as Europe, Central America or South America, the large population and geographical distances can absorb such a biological attack. On an island, even a relatively large one such as Cuba, the smaller Indian population had no place to run from the diseases and were wiped out.
As Luis pointed out, the Indians were not wiped out in Spanish territories of what is now Mexico. In fact, the Indian gene pool was and is the major component of what Chicanos call "La Raza" (The Race).
As a result, the Mexican views himself as the descendent of the original Indian inhabitants and views both Spaniards and the Americans as the hated Euro-invaders who stole their land.
This hatred of America is what I see as the major problem in current immigration be it from Central America or from the Middle East.
In the past century and a half, millions of immigrants may have come to the U.S. more for economical reasons rather than political reasons but, for the most part, those immigrants did not hate America and it's institutions. Nowadays, however, the U.S.A. has opened it's doors wide open to millions of individuals that seek American economic benefits but hate America and it's values.
There is no problem with "diversity" as long as diversity means being a Creole-speaking Cajun, Black, Southern, Boston Irish, Pennsylvania Dutch, Hawaiian, Cubano or Heinz 57 as long as the common bond is a true love for America, the Constitution and the core values that made America great.
However, if "diversity" means that every America-hating ethnic group in the Planet can come in by the millions and set up housekeeping in the U.S., then, eventually, the philosophical character of America will change.
Once America loses the core values that made it great, America will die just as the Roman Republic died.