I am a native citizen of the USA.
Is it incorrect to refer to the country as America?
It depends on context. Within the USA, it is generally understood that "America" refers to the USA (whereas "the Americas", for example, refers to the combined continents of North and South America). Note: This isn't really because of "narrow-mindedness" or anything like that, it's simply because, over time, it has become a norm in AE to use it as shorthand.
Outside of the USA, it depends on both where you are and the context of the conversation. For example, the use of "America" in other countries in the Americas may be offensive to some, or at least ambiguous. Outside of the Americas, probably not offensive (unless you are in a country closely tied to a country in the Americas), but certainly with the rare possibility of ambiguity.
And even if it is, won't people from around the world understand that the USA = America?
As mentioned above, it really depends. There is no global rule, and the interpretation will depend on where you are, what the common usage of "America" is in the country you are in, and the personal beliefs of whoever you are talking to.
If you are concerned or unsure, simply say "United States". While that is also technically ambiguous (there are other states that are united, actually the official name for Mexico translates to the "United Mexican States"; but for whatever reason it has become more commonly known as "Mexico"), it is commonly understood to mean the USA (even our government uses this shorthand globally, e.g. at the customs office you will see "United States Customs"), and it won't conflict with any other persons' use of "America". Additionally, within the USA, "United States" is commonly used and understood and won't raise any eyebrows or cause any confusion; it's a good global safe bet.
If you're really concerned, "USA" is 100% safe in and out of the country.
And, of course, within the USA, you can always say "America" without issue.
As Marcel Turing noted in the question comments, it isn't really an issue of "correctness" as much as it is an issue of common usage. Of course, in reality, it's slightly more complex, because while in an ideal world people wouldn't take offense at things other people innocently say, that isn't how it works. So you always need to at least be aware of local attitudes (and again, when in doubt, just go with "USA").
Personally, when I'm home I say "America", and when I'm travelling (even to e.g. Canada) I say "United States". This has never caused an issue (even in Mexico).
Edit: As for the demonym, as ntoskrnl points out in the question comments, "American" is the official one (but is shared with the continental demonym) and is usually understood, depending on the context, as referring to the country (with the same caveats as above; and unless the context is continents) with the only other real global option being "US Citizen". Wikipedia has a good article on Names for US Citizens. It also gives a nice overview of "American" in Demonyms: Cultural Problems.
The adjectival form has essentially the same issues, but without the "US Citizen" option.
ChrisW's answer gives a nice overview of context.
Please note this is all based mostly on personal experience, not so much canonical reference.