As Hawkeye stated, the USCIS prefers the term U.S. citizen. Quoting directly from my United States Certificate of Naturalization, I am referred to in different instances as a "citizen of the United States" and a "citizen of the United States of America". My U.S. passport also lists my nationality as "United States of America".
But obviously these documents are written in a formal register. When I naturalized I also received a ceremonial letter signed by Barack Obama which opens "Dear Fellow American". There's no reason it couldn't have opened "Dear Fellow Citizen of the United States" except that it sounds awkwardly distant for a letter with a friendly and welcoming tone. One could make the rather unhelpful and misdirected argument that because "America" describes two continents, it should only be correctly interpreted as such, but the meaning of the opening I cited is clear given the context and in no way unusual.
I think that these nuances are only really important in international contexts where the intended meaning may be cloudy... except that "American" as a convenient alternative to "U.S. citizen" is universal enough in English that, though it may invite some resentment, it really isn't cloudy as long as there is no popular awareness among the Anglosphere of some hypothetical Canadians, Mexicans, and others throughout the American continents adamant about a preference for being considered "Americans" in extranational English language conversation.
This is different and separate from the England / Great Britain / United Kingdom issue, primarily because the distinctions between these terms are of actual legal and social significance, and are upheld by sizable populations within the region.
Imagine that I'm from Vermont and I tell somebody I'm from New England and they respond "Oh, you mean Massachusetts, yeah, I've heard of that place before"; they are wrong because they have mistakenly or in ignorance taken Massachusetts to mean all of New England, or perhaps think New England is an alternative name for Massachusetts. In either case, this conversation partner clearly doesn't understand where I'm from (I could be from Massachusetts, but not necessarily). This is the equivalent of people misunderstanding United Kingdom to mean England (much to Raven's chagrin).
If I instead tell someone I'm from the U.S. and they respond "Right, America", in most all cases I would feel secure in the knowledge that they haven't completely forgotten their geography because America is so widely understood to mean the United States. It also helps that, as Balcerzak said, there is no continent or other country called simply "America".
I don't think there are any problems with America being used as a casual name for the United States (though I wouldn't oppose any movement to change that, and I personally prefer to avoid the term "America" and its derivatives in my writing). I do think it's problematic when people take England, Great Britain, and United Kingdom to be interchangeable because when they do it's often unclear what they mean to refer to (because they are all distinct entities), but I'm British so admittedly I'm biased.