US-ians are typically referred to as Americans...
As for the continents themselves, it just depends on who's doing the teaching. English speaking countries define North and South America as separate continents:
The bit I was digging into there, by inference (not necessarily clear for those not familiar with Spanish) is that because North and South America are taught to be one continent in Spanish, and "americano" references the peoples from that aggregate, Spanish speakers (mostly Latin-Americans) have a habit of correcting English speakers in online forms when they use the word "American" to refer to people from the US.