Can the Doctor be something other than a white male? Yes
Should the Doctor be something other than a white male? Maybe
Sure, the Doctor could be something other than a white male. The question becomes "Why is the Doctor something other than a White Male"? From what we've seen of Gallifrey, White Males are in charge there too, so it's more than just "Britain is 87-91-74-98-104-1.01% White!" (your numbers may vary depending on how much of a nitpicky flame war you want to engage in)
Given what we currently know about the entire situation, there doesn't seem to be a compelling reason to change the Doctor to something other than a White Male.
This isn't to say that the Doctor should always be a white male, or can never be a non-white male. As a rather appropriate example, see: Dr. Joan Watson. Baby step, true, but still a step toward a more inclusive Sherlock Holmes. Elementary is set in a significantly different setting than traditional Sherlock Holmes stories and so making a radical shift in one character - both gender and ethnicity - which doesn't actually negate the character at all. Still a Doctor, still assisting Holmes, still extremely competent. Just now an asian woman instead of a white man. (And yes, there is a better example to use from this show but the spoiler tags would give it away.)
Can Doctor Who do this? Yes. It might even be able to do it without radically changing the setting, as Elementary did which allowed it to make these kinds of changes. But should Doctor Who do it? I'm not seeing a compelling reason why it should. It's not likely to lose fans like me - newcomers who aren't all that invested in the history of Doctor Who (particularly Doctors 1-8), as long as the show is still entertaining and fun. It might turn off long-time Doctor Who fans who have spent 50 years with a white-male Doctor and don't want to change. It might trigger some unconscious resentment or lack of respect for the new Doctor. (there's a few studies I've seen referenced about how Americans react to different ethnicities in positions of power which indicated that Americans tended to react best to white males being in charge. One of them had to do with looking a pictures of people and told to pick out the people they thought were most powerful; 6 headshots of random men and women and asking the participants to point out the "Police Officer" or "teacher" or "Lawyer" or "Doctor", etc. The higher the position of power, the more often participants would point to a white male in the photo line up. And older white males, I believe, were highest on the list. My Google-fu is weak tonight, though, so I haven't got the reference handy. And yes, it was about Americans, not Canadians or Brits - though I seriously doubt that the difference in this case would be significant.)
Or it might just be that the suits who market the show and the writers who write the show can't be arsed to change; they've got a good thing going as it is, why change it? Why try to change the marketing of Doctor Who to revolve around a Woman and her male Companions? Why write out how Queen Elizabeth would react to her husband (David Tennant) showing up a few days after they get married... as Chiwetel Ejiofor? Could just be laziness on the part of a long established and well loved show. Who knows?
Overall, I'd be quite interested in seeing a non-white, non-male Doctor, or maybe a Transgender Doctor of some kind (Cross dressing doctor, for example) in the future. But I'm not the shows bread and butter; I don't go to conventions dressed as a character from the show, or even buy the DVD's (which are ridiculously expensive, by the way). I'm a pretty casual fan who hasn't invested a great deal of time, so a potentially major change like this wouldn't rock my world.
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