With exploration, oddly enough, it usually isn't a single "someone" who makes such a critical realization. Especially in this instance, the chances of there being a person with Columbus who had actually visited the eastern edge of Asia (where Columbus thought he had arrived) were very slim. Trade for Asian goods was largely conducted through North Africa or overland, and the only overseas route was dominated by Portugal, who didn't travel very far east (they mainly stayed in India and near Malaysia).

The primary confounding factor was the primitive state of geography. It was only just developing into what you could call a science, and at that point was really more of a tradition or art. Columbus made a pretty reasonable and precise computation, he just had incorrect data and some false assumptions. At this point there was no good way to calculate one's longitude, making the determination of your east-west position on the globe that much more difficult.

It didn't take long for the realization to happen, however. Portugal was pretty suspicious from the beginning, and actually "captured" Columbus on his way back from the first voyage. John II immediately questioned Columbus' conclusions, but that was more motivated by his vested interest Spain having failed to find Asia than by legitimate scientific skepticism. This isn't to say that Spain was confident in Columbus' assertion. They avoided claiming a new route to Asia from the beginning.

Columbus' final voyage was pretty important in that he reached Central America. After he died, further Spanish and Portuguese expeditions revealed the existence of South America, and much of the Brazilian coast was charted. This was aided in large part by Vespucci, who had more accurate data and instruments at his disposal. In around 1503 you start to see references by Vespucci and map-makers to a "new world" or a "fourth part" of the globe (in addition to Europe, Africa, and Asia). By 1520, Magellan sailed around South America on his way to circumnavigation.

The islands were referred to as the "Indies" and the people as "Indians" for a long time primarily because of convenience and convention. They were called this in official documents by Spain, probably for practical or political purposes.

Short answer: There was a lot of serious doubt immediately after Columbus' very first voyage. Within ten years, people became confident that the new lands were not in Asia. Within thirty years, they had sailed around the new continent. The names stuck because of social and political convenience (it became a sort of meme).

Sources:

Gibson, Charles Spain in America

Parry, J.H. The Age of Reconnaissance