False. In Latin America, the English word "America" refers to the USA. The Spanish word "América" refers to the entire continent. Nobody will ever confuse the two because you're speaking two entirely different languages- or if you're speaking Spanglish, EE.UU. is available, as is the demonym "estadounidense", whereas there is no "United Statesian" in English. The English demonym is "American" for US citizens, just like the English demonym for Canadian citizens is "Canadian", and Guyanan "Guyanese".

I don't know why Americans feel offended when they are called Gringos.

Because it's used as an ethnic slur? I dunno what to tell you man, but by the same reasoning "nigger" isn't a slur because it's simply an observation of someone's skin color and isn't meant to offend.

Part of the problem is the cultural distance in racial politics and political correctness between the contemporary spanish-speaking world and the contemporary english-speaking world, where in the latter, overt reference to the national origin or race of someone is inherently suspect and is avoided with circumlocution or euphemism. See, for example, Downton Abbey where every time the Dowager countess makes comments about "foreigners" (already a euphemism) people get upset.

You may disagree with this, but that's just the way English is spoken here. While I may not agree that using something Americans perceive as an ethnic slur as an every-day term for Americans is a particularly good idea if you don't want to offend Americans, I understand that is the way Spanish is spoken in most countries, and Spain in particular has a culture of extreme resistance to changing the words they use in response to foreigners being upset.