There really is no actual reason why the Doctor cannot be anything other than white or male. It is just convention that suggests that he always has been and therefore should always be so.
I suspect that a lot of pedantic fans out there will cite the fact that there are time ladies which would suggest that there are two distinct genders amongst Time Lord’s or Gallifreyan’s (depending on your point of view) but, after it was stated in the television series that there was a Time Lord (the Corsair), who had both male and female incarnations then there is no reason to suggest that they can’t.
I am not one of those people who would be bothered if we had a non-Caucasian female in the role of the Doctor, as it wouldn’t really change the character itself as the Doctor is always the Doctor no matter what kind of body they are in.
It might affect the way the character is seen, and perceived, by other characters depending on the stories setting such as in the historical stories but for any set on an alien planet for far in future I cannot see why it would made a blind bit of difference what the Doctor looks like.
It wouldn’t phase me, and to be honest it would quite refreshing if we had a totally different type of Doctor to the others, and as long as they are right for the role then it doesn’t matter what sex, colour, shape they are.
As for what sort of female I would like to see in the role, well, that’s another story…