Why name the Americas "America?"

America is named after Amerigo Vespucci, the Italian explorer. He is the man who set forth what was then considered to be a revolutionary concept: that the lands that Christopher Columbus sailed to in 1492 (but didn't actually land on) were part of a separate continent. 

A map created in 1507 by Martin Waldseemüller was the first to depict this new continent with the name "America," a Latinized version of "Amerigo."

Detail of the Waldseemüller map showing the name "America."
The Library of Congress purchased of the only known extant copy of this map
for $10 million, thanks to the generosity of the U.S. Congress, Discovery Channel, Gerald Lenfest, David Koch and several other donors.

Waldseemüller's large world map included map data that had been gathered by Vespucci during his voyages of 1501-1502 to the New World. he named the new lands "America" on his 1507 map in the recognition of Vespucci's understanding that a new continent had been discovered.

See www.loc.gov/wiseguide/aug03/america.html