Yes, but America is not a reference to a continent. That would be North America or South America. (I suppose you could call the super continent just "America" but most call it "The Americas" instead.)
Clearly the term "America" either disambiguates to a continent or a country, and most assume the country.
"Europe," on the other hand, really is a continent, so the term has a completely different meaning than the EU, although it would be reasonable for the two to be synonymous eventually, they clearly are not currently.
Anyway, it seems reasonable to use the term America as shorthand, instead of U.S.A., as it's pretty obvious what you mean.
And in the description of the United States of America:
The United States of America (USA), commonly referred to as the United States (U.S.) or America