No Native civilizations =/= no Native peoples.
When I (and from what I read, OP) use the term "civilization", it refers to a certain kind of society which has the following characteristics:
Large-tens of thousands, hundreds of thousands or millions of people.
Hierarchical-There is a king/president/noble class which can order around people not in that class.
Specialized-The remainder of the people are divided into farming, artisan, merchant classes etc.
Organized-Information is stored physically instead of just purely orally, using techniques like writing or
quipu. Taxes are gathered from the population and given to the elites or redistributed, and workers can be gathered (sometimes against their will) for large projects like monument-building or waging war.
Civilization is not the only kind of society. The Yahi of California lacked the numbers, organization, and specialization of the Aztecs. All members of the tribe gathered food AND made art AND traded as necessary, and all the men fought as needed. But just because their society was not an example of civilization does not mean it was less valuable (in fact, quite a few people would argue that their society was better), and it is a tragedy that they were wiped out by white settlers in California. The Aztecs, on the other hand, were conquered but due to their larger numbers and the fact that they had a government and social structures their conquerors could exploit were not wiped out. Members of the Nahuatl ethnic groups that constituted this empire are still around, and their culture also survives in a recognizable form in Mexico's Mestizo culture. While they suffered conquest, they survived genocide but the Yahi did not.
The settlers of Uruguay and California didn't wipe out any civilizations, but they did wipe out many societies and the thought of this happening on a larger scale is sad for anyone who values human life.