Will an actress ever audition, though? Will a showrunner ever audition a woman for the role? That doesn't guarantee them it, for starters. If I may point your attention to the point in time where Paterson Joseph seemed to be the front runner for the Eleventh Doctor, only for the press release to read 'the youngest ever actor' rather than 'the first black actor to play the role'.
While no one running the show is going to ignore an actor because of their skin colour, a woman has yet to be publicly revealed as ever having auditioned. A person's sex is part of their identity, as is their skin colour. Whether or not it should be is another debate.
Outside of the story universe, how much detail do you have to go into to explain to a child the differences between a hypothetical female Twelfth Doctor and Matt Smith? Would children cope? Children would probably be more willing to accept it than adults, because they're less familiar with the history of the character. Would adults cope? They're the ones who might have to get into overly-technical explanations if they are blessed with sufficiently inquisitive children. Then again, would it be any harder to explain than Jenny and Madame Vastra?
Explaining that he looks different should be easy, because he always does look different. Some children stopped watching when Eccleston became Tennant. Other children started. Audiences come and go with change, but would this risk losing too many viewers? Explaining that he is now a she might seem to have more potential problems than explaining another man in the same role, but we simply don't know. However, the real-world arguments against a female Doctor seem more potent to me than the in-story ones.
So far, the Doctor has been played by white men. Even alternative versions of him have been nearly entirely white (unless you count Daniel Anthony playing the Eleventh Doctor in Clyde Langer's body). 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).
This trope is ingrained. It is hard to shift. In medical circles for years 'The Doctor' was associated with men, as 'Nurse' was with women. That stigma still exists today.
In reality, of course, that's simply not true. As society has changed, so too has the character of the Doctor. Some incarnations have been more like the lapsed academic than others, but since 2005 Matt Smith was the first incarnation to really exhibit this side of the character. Russell T Davies created more modern Doctors in a universe without the Time Lords. Eccleston was bullish but damaged. Tennant was all quicksilver synapses. Of course it would be possible for a woman to have the characteristics which have worked so well for the character.
If you prefer the Middle-Class-White-Man version, well, do you not know any women who resemble this character? Miranda Hart has become incredibly popular playing the female version of this in her sitcom. It exists. And of course, actors play other people. Character actors have been successful in the role before. The actor doesn't have to exhibit these qualities in their personality.
Joanna Lumley played the Thirteenth Doctor in Comic Relief's The Curse of Fatal Death, and Arabella Weir played the Doctor in the alternate universe, 'Unbound' Big Finish audioplay Exile. This is it, so far. Two performances outside of canon, one of which lasted for about thirty seconds and the other written deliberately to be different. After years of speculation, jokes, and spin-off material postulating, it has been confirmed on screen (most recently in reference to The Corsair in The Doctor's Wife, The Brilliant Book of Doctor Who 2012, and Shada) that Time Lords can change sex when they regenerate. The story universe would allow for this change to happen if it came to pass.
At the end of the day, it is up to the people who run the show whom they cast. If they audition a woman, though, and they don't give her the role, there might well be a wider outburst of the undercurrents of discontent after the last few major casting announcements: all white people. Is it prejudice, or is it because they were the best people for the role? There have been disparaging comments saying 'Oh wow, another white person, what a shock', which brings forward the question of positive discrimination.
Looking back, any initial disappointment or anger concerning Matt Smith winning the role over any black actors largely abated about three seconds after he said 'Can I have an apple?' If they cast a woman, is the risk any bigger than the casting of Matt Smith? If we recall, the pressure was huge for him, and he won an audience over. Would the pressure really be any greater? It can be done again by any other actor or actress of sufficient quality.
If the show remains popular, then the ethnicity of the leads won't be questioned by anyone other than a minority. If the casting is right, then the actor will be popular enough to convince the audience. Personally, I don't have any problem with an actor of any age, sex, creed or colour playing the Doctor if they're good, but I doubt a female Doctor will ever happen, because the character has been male for so long, and because it doesn't need to happen.
The show, for all its flexibility, has a formula that works, and just because a woman playing the role could work, doesn't mean it we'll ever get to find out if it does.
