Andrew Blair, writing at Den of Geek, discusses the gender and ethnicity of the Doctor:
So far, the Doctor has been played by white men. […] There is a strong argument to say that the character is simply a man, because he has always has been. He’s just…mannish. Professorial. Or, looking at the history of Gallifrey and the Time Lords, he’s quite possibly the embodiment of Middle-Class White Man Guilt.
It’s certainly a valid interpretation: he gets bored with his life on a planet of dusty academics and administrators, having graduated from the Time Lord Academy. He then runs away from it all to explore the societies considered beneath his race, and overcomes his own snobberies as he goes. He would love to fit in properly on Earth, but he can’t quite manage it. This is a very British creation. A post-colonial alien race based on the British upper-classes. The Doctor is an Oxbridge graduate allegory, fleeing from what is expected of him by traditional values, ashamed of his lack of real world knowledge.
It’s a staple role in British satire (think Wodehouse and Waugh, Douglas Adams and Monty Python). The character is traditionally male, the brilliant but insecure middle class man who gets easily bored and distracted. It’s part of the essence of the character (the main gist of the argument is that he has to be male, even if you disagree with this idea of the character).
Yet, as Blair points out, there are ways for the image of the Doctor to be expanded and subverted, whether it be a make or female actor, black or white actor, or anyone.