NOT A TEACHER
This is not a case of how non-natives should refer the the United States of America. If it is clear from the context that "States" refers to the USA, then sure, by all means, do use "States". It's very common. However, if you want to be specific and avoid ambiguity, then "America", " the US" or "the USA" would be more appropriate. This is just my opinion. "States" can refer to a number of things, but it often means the "USA". However, I usually prefer to avoid ambiguties and I would rather use the "US" or the "USA".
Edit: the Longman Dictionary of Contemporary English says that "States" is an informal way to refer to the USA:
States, the / steɪts /
an informal name for the united states of america