Is a white person born in South Africa who moves to the U.S. an African American?

LOL That IS the question!  Not until he BECOMES an American citizen, anyway.

if a person is born in South Africa, or anywhere else, to a U.S. citizen who is abroad, that person is a U.S. citizen at birth, and would remain so after his return.

Once they get US citizenship. 

I used to work with a Dr from S, Africa, who, the day after her naturalization,  walked into work and said "I am know officially an African American"

A white South African who moves to America remains a South African... you don't become an American just by moving.

No they're a South African. Their nationality doesnt change.

Yes, and no.

Language is an imprecise art. Yes, an art, not a science.

In the context of the USA, an "African American" is understood to be a black American with African ancestry.

A white South African who becomes a US citizen?

Yes, definitely, they are from Africa.

And yes, they are now American.

So... an African American? Historically and contextually, that description applies to a distinct segment of American culture, black society, who are bound by a history of oppression, and a shared social context in modern society.

Bear with me, I'm a white Australian guy trying to navigate this minefield as best I can...

A white South African expat is a very different individual from those that are generally known as "African Americans" in the context of modern America. The term "African American" has meaning in an American context that does not apply to a white person from South Africa.

That doesn't mean that they can't be friends, or maybe play Counter Strike together sometimes...

Just like we have Italian Americans, or Irish Americans, or whatever, any white person from South Africa would have European heritage.  White is not an ethnic group, just a skin color.  So if a white person who's family came from England were to be born in South Africa, then they would be English African.  If they moved to the US then technically they would still be of English heritage and become an English American.  Every white South African I have met though, are just referred to as South African.  People know that a white person from South Africa has European heritage.