Vespucci was the first person to identify the American land-mass as a new continent (Columbus thought it was the Asian coast). But it was a German cartographer, Waldesmueller, who was the first to draw a world-map showing the new continent discovered by Vespucci. Waldesmueller named that new continent "America", after the Latinization of Vespucci's first name. The latinization of Vespucci's name is Americus Vespucius. Hence America, and the name stuck.
Vespucci was a very technically expert navigator. I read that he knew how to determine longitude by the Lunar method, a difficult method that very few people could use (longitude wasn't easy to determine until Harrison invented the marine chronometer.)
Vespucci's ability to determine his longitude is probably the reason why he knew that he wasn't at the Asian coast, and was at a new continent. Columbus didn't know that, and, as I said, Columbus thought that he was at the Asian coast.
So: Vespucci was the first European (if you don't count the Vikings in Laborfador--but why wouldn't you count then??) to identify the new continent. Of course the first discoverers of the new continent were the migrants from Asia who later have become known as the Native-Americans.
And Waldesmueller was the first person to make a world map showing the new continent. ...and gave it its current name.
Incidentally, it seems to me that the map-projection that Waldesmueler used was one of Ptolemy's map-projections, or the (now called) Bonne Projection, which differed from Ptolemy's map only by equally-dividing each parallel with its meridian-crossings (something that didn't matter in Ptolemy's time, when no one could determine longitude anyway).