I can honestly see absolutely no justification for having a problem with the doctor being non-white other than ... well ... racism, sorry. I'd say it's FAIRLY obvious that, say Chris Eccleston and David Tennant have different genetic histories, they're not similar-looking men, so in terms of the core concepts of the show there's no reason he'd constantly be a white man.
I have more sympathy for the notion that The Doctor shouldn't have been made female, simply because it makes more sense (in terms of genetics) that Time Lords would have a biological sex and each regeneration sould still be the same sex. That said - it's high-concept sci fi so

if it's established lore, it's established lore. I also have to give Doctor Who in general props for always establishing precedent; we saw a time lord swap genders at regeneration with Missy. We've also seen time lords regenerate with a race change before.
So long as they don't fundamentally change the character there's no issue. Or there shouldn't be. Which is why I'd find it hard to swallow if The Doctor suddenly became exclusively gay. The race "issue" should not be an issue.