The first time the name "America" was used in a significant manner was on the "Waldseemüller-Map". This wall map made by the German cartographer Martin Waldseemüller was properly called "Universalis cosmographia secundum Ptholomaei traditionem et Americi Vespucii aliorumque lustrationes" - The Universal Cosmography according to the Tradition of Ptolemy and the Discoveries of Amerigo Vespucci and others.

Amerigo Vespucci was, most likely, chosen as a namesake by Waldseemüller because Vespucci was the first navigator not to claim that the Americas are part of India, but a landmass of their own right. And as you can see on the map, Waldseemüller depicted the Americas as a separate landmass, and in line with that labeled the landmass "America".

As a follow-up, why was America not named "Vespuccia" (or something similar) considering most other places named after explorers use their last names?

Christoffel Columbus refused to believe he didn't reach India, even on his deathbed. Vespucci was the first person to state that the landmass Columbus had reached was in fact an unexplored continent. Even though he explored a lot of the New World, his attribution to it has been exaggerated.

He (Vespucci) sent out letter which became widespread and caused people to call the new continent America. When Martin Waldseemüller made the Waldseemüllermap, he also called the continent America.

I've heard several mentions of a man named Richard Ameryk and it being named after him. Any truth behind that?