There has only been one single place on this planet that has ever been known as "America", and that is the United States. There's the term "The Americas" which is synonymous with saying "The New World", and can be used to refer to North and South America together, but the only entity that has ever gone by "America" has been The United States.
There is a demonym for the people who come from or are associated with all six of the inhabited continents: Africa - African, Asia - Asian, Australia - Australian, Europe - European, North America - North American, South America - South American. That ought to be simple enough to understand. The term "American" used as an adjective and as a nationality, refers specifically to people and things that come from or are associated with The United States. This is especially true for English speakers. Most Australians, Britons, Irish, New Zealanders and South Africans refer to The United States as "America" in common speech, as do Americans themselves, and have been doing so for a very long time.
The Oxford English dictionary defines "America" as "a name used for The United States" and "American" as "Relating to or characteristic of the United States or its inhabitants." Secondary definitions acknowledge that the noun can be used to refer to the entire land mass and that the adjective can be used to refer to any person or thing associated with it, but few people - and practically no one who speaks English as a first language, would do so.
The plain and simple fact is that "America" and "American" are well-entrenched terms for the USA and her citizens. And while there may well be some segment of the world's population who takes issue with that fact, it's not going to change anything. In every official capacity the inhabitants of The United States are referred to as "Americans", not as "United Statesians."
The main argument against this logic nearly always comes from Canadians who protest that they too ought to be considered "Americans" because the land mass they call home happens to be situated on North America. If they wish to call themselves "North American", then they are free to do so, but The United States appropriated the term a long time ago, it has since become common nomenclature the world over and it's not something that's going to change anytime soon, and probably never will.
The United States was the first independent country in the New World, which is a big distinction. Therefore, the people of the USA had first dibs on the name, and they succeeded in securing it for themselves. Whether or not the rest of the world wishes to recognise the exclusivity of the name is irrelevant, because it's those who refuse to accept the fact that it's become common usage who are bound to run into problems, not Americans themselves.
In Ireland, The United States is almost always referred to as "America." And we adopted that custom from the British who have been using it from the very beginning. It would be a lot easier for all parties involved if the people who take umbrage with the use of the term were to simply come out and say "I have an axe to grind with the USA and this is my petty attempt to disguise my envy", because that's all it really is. I've personally seen Canadians laughed right out of pubs for getting their knickers in a twist over this in Britain and Ireland where nobody blinks when the terms "America" and "American" are used.