No. She doesn't. Matter of fact, according to one source, Whoopi's family background is entirely African. Not that would stop her from converting.
But she hasn't. The surname is said to have been at the suggestion of her mother who felt that Johnson wasn't Jewish-sounding enough to thrive in a Jewish-dominated industry.
Frankly as a Jew I take that as a great compliment.
Ironically, Goldberg says that people often ask her if she's Jewish. She replies: "Would you ask me that if I were white?" (Yeah, I would, because anyone with that surname might have a Jewish father & gentile mother, which means you aren't Jewish). The irony to that is, she isn't.
I take her comment as historically incorrect, but not offensive in the least. She applied an American definition of race to Jews, which does leave, most of us at least, white.
So, in fact, bravo to her for not leaving it to the Nazis to define who is what.