There's a reason the Americas are called that, rather than the Columbias. To his dying day in 1506, Columbus insisted that the lands he reached were still newly discovered (because they had to be new in order for him to be governor and get a cut of the profits Spain would reap), but part of Asia, based off his entirely incorrect calculations about the Earth's circumference.

Instead, the man who determined that these new lands were in fact entirely new lands was Florentine merchant and explorer Amerigo Vespucci, who chartered at least two, possibly as many as four expeditions to the new world. Vespucci may have been the author of letters to Lorenzo de Medici and Piero Soderini, both of Florence, describing what may have been some of his voyages to the new world, and how during those voyages he sufficiently mapped the coast of the landmass we now call South America to be certain that it could not be Asia.

You may have noticed that the previous paragraph sounded weaselly, with lots of "may have" and "as many as". The hard facts tell us that Vespucci was part of two expeditions, in 1499 and 1501. Less certain are the expedition before these two in 1497, and after in 1503. Only the letter to Soderini mentions the first and fourth expeditions, and there is some evidence to suggest this letter was a fake written years after the fact due to inconsistencies in the narrative and with other letters written by Vespucci. The often suggested reason for this would be that, if Vespucci's first expedition did happen, and was exactly as his letter suggested, then he was the first European to set foot on the new continent, beating Columbus by over a year (and obviously having been beaten by half a millenium).

I'm not remotely qualified to judge the letters' authenticity, nor is it relevant to this discussion, they were at least believed to be authentic at the time. Vespucci's letter to de Medici describing his third voyage was published and distributed around Europe starting in 1503, first in Latin under the title Mundus Novus (New World), later in other languages. Like the Soderini letter, Mundus Novus doesn't match up with Vespucci's style of writing and gets some details wrong, however, the general assumption I've found is that either de Medici himself or those hired to publish it rewrote the letter in an effort to punch it up, rather than being entirely fabricated.

The letter to Soderini, which didn't get a fancy title and is simply known as the Soderini Letter, was published in Florence in 1505 and came to the attention of a group of scholars in France a year later, when they managed to get a translation of the letter and a map detailing the coasts the expedition covered. They believed the new continent to be the opposite land theorized by Roman scholars as early as Pomponius in 43AD, and even conjectured by Plato, a land where a person standing upright would be upside-down compared to a person standing upright in Europe, sometimes called the Antipode (a word which now refers to two points on Earth completely opposite from each other, and briefly referred to the people living on such a theorized opposite land, rather than the land itself).

That group of scholars in France included the German cartographer Martin Waldseemüller, who used the information from the Soderini letter and accompanying maritime maps to create the earliest map of both the old and new worlds, (warning, this image is massive) obviously woefully incomplete, but at the time the most complete map of the world created. Waldseemuller's map and globe with the same new coasts added to it were made to accompany a book written by Waldseemuller's colleague, Matthias Ringmann, and it was in this text that Ringmann proposed calling the new landmass America (with Waldseemuller having already done so on the map and globe), noting:

"I see no reason why anyone could properly disapprove of a name derived from that of Amerigo, the discoverer, a man of sagacious genius. A suitable form would be Amerige, meaning Land of Amerigo, or America, since Europe and Asia have received women's names."