Yes, but that’s part of the point: in the most formal registers, “America” is incorrect-if you’re a lawyer drafting a legal contract, or an academic writing an article for a peer-reviewed journal on international relations, you’d be much more likely to write “the United States” (or “the US” for short) than “America”-and if despite that you wrote the second rather than the first, it is likely someone else would “correct” it in the editorial process

So there is a very real sense in which “the United States” is more formally correct than “America”. But it is of course context-dependent: using the most formally correct term is likely pragmatically incorrect if your audience is a classroom of average seven year olds

Also, formality of speech isn’t just determined by context (a legal contract or a peer-reviewed article versus speaking to a primary school class)-it is also determined by the background of the speaker (and audience)-people who come from more educated/professional/higher SES backgrounds tend to speak more formally even when speaking informally; the same is true of higher IQ people and higher AQ people (AQ=autism quotient, measuring autistic traits)