Oh this is silly.
Of course Americans know you are white. However we Americans are SO programmed to pretend we do not see color so as to seek any label but the obvious one, for fear of offending, or being labeled a "racist".
A white man is not a white man...to speak of such is equated unfortunately to certain racist groups portrayed on the media...likewise many shy away from calling a black man black, for fear of causing offense.
IMHO it is ridiculous. Black people know they are black, white people know they are white, yet we play these silly hyphenation games. I'm not black...I'm African-American
. I'm not "white", I'm Caucasian
using other labels is even considered by some to be "insensitive". Hence referring to people by national origin, rather than, heaven forbid, the race that anyone can see.
Political correctness run amok,
and I wouldn't be surprised if Americans on all sides of the issue were tired of it.
Yet, it is how things are.