Actually no. Society determines race, not genetics. If you have all white-people genes but for some reason look black (absurd, this is the part genetics plays... but I'm exaggerating to make the point), are raised around black people and believe yourself to be black... with other people also thinking you're black... you're black. Race is completely about identity based on physical characteristics. In day to day life we do not examine people on a genetic level to determine what race they are. We go by what they look like and who they identify with. There is nothing particularly special about the arbitrary characteristics of races that deserves classification... simply that we see difference.
So a part-black person who grows up in a black neighborhood and who looks more black than white- He's black. He's also of mixed race. Kind of like how I have a little Apache blood... but I'm white. It would be stupid for me to call myself mixed-race (though I do tan pretty well for someone who's usually pale

).
It's also ludicrous to think that his status as being part-white changes anything. Mixed race people are often more socially stigmatized because they are not seen as members of either group. Thus when trying to gain the sympathies of the "white liberal establishment", mixed race works just as well.
I'm not exactly impressed by the website this story is on. It's obviously biased and leaves much to be asked for in the way of evidence that Obama is truly hiding anyone. Since when does anyone give a flying %%%@@% about the grandparents of politicians? What do we know? Do we know what the current health condition is of both grandmothers? Do we know what their relationships were like in recent past? Do we know where they each live in relation to Obama's permanent residence or Washington?
None of that has been satisfactorily established. All we have is the assumption of conspiracy for something so dumb as whether the politician shows off his grandparents in an equitable fashion!
It all seems really ludicrous when you consider from the beginning any one who paid even the slightest bit of attention knew that Obama was part-white. It's not a secret. So what? Are we supposed to be shocked that his grandmother on his white parent's side is white? Christ! What a shock!