As a black woman, I see Obama as a black or African-American (although I am not a fan of that term). Saying that he is not really black because his ancestors were not American slaves is like saying that someone isn't a real white American unless their ancestors came here on the Mayflower. It doesn't make sense. If he looks black, then he is black. Now, he doesn't have the same heritage as most black people in this country, but that is a different issue.
I never understood the obsession with the abundance of melanin in determining ones "Blackness". I sure hope, and wish, the issue had logical value.