No, it's not right. A Korean American would be someone who lives in the US (as a citizen) who is of Korean heritage/ethnicity. That could be an immigrant himself, or someone born here.
To the extent that a person or his children become assimilated, they would begin to identify simply as "American."
If such a thing as an "American Korean" existed it would be someone who was a citizen of Korea, but of American ethnicity. (Which is a problem because "American" isn't really an ethnicity.)
Student or Learner