I doubt many Spanish or Portuguese speakers refer to themselves in English.

Regardless, sure South Americans can absolutely call themselves Americano in the continental sense. But I know in Brazil for example "Americano" is casually understood to mean from the US, and in general South Americans are more likely to identify as argentino, brasileiro, chileno, colombiano, etc., or as sul-americano/sudamericano.

Most importantly, when speaking English, virtually all will avoid American for themselves because they know in English it means estadounidense.