Why would it be "short" for the United States of America? The reason they call it that is because it is a country comprised of many states formed into one, and they are within the landmasses of North and South America, which together they are called America or the Americas. It really does not make sense to call the country the same thing that the two continents are called when viewed together! I may be ignoring what this loving world calls this nation, but at least I'm being logical about it.
Well actually both continents together is The Americas, also there isn't any reason people would ever use the word America except to refer to the United States of America, as each content has its own name that is used, North America and South America. It is actually illogical to have a word for both continents, I mean there isn't a word to refer to Europe and Africa at the same time, when you want to talk about both you just say both.
Here is what would be logical:
North America refers to North America
America refers to United States of America
American refers to people from United States
South America refers to South America
Canadian refers to someone Canadian
Mexican refers to someone Mexican