I kinda went into thinking about the female doctor thing in the same way as a lot of people, that it would be dumb, he's a he, always been a he and should stay a he.
Thing is as I thought about it, arguing the point, the stuff I started to cite in that argument made me change my mind.
To start off I hate the current repackaging of properties, switching genders, its quite often being done for a gimmick. Its quite insulting and a cheap trick to cash in on an established character/property for the sake of it and more realistically the money. I would much prefer it if instead of this 'reimagining' they created new, strong female characters and stories, instead of simply looting existing male ones. I don't think its a very good example and is kind of insulting to women. Its essentially saying they would be good if they were men really. The resistance is somewhat fair in that sense, there would be a lot of backlash, maybe even worse, if female characters were being switched in the same way.
That said, in all of popular fiction/culture, the Doctor is maybe the one instance were it would work and actually make sense.
As it was originally conceived (and produced by a woman) Doctor Who can be about anything, any time, any place, any kind of story. A mystery, a whodunit, a war, a romance, fantasy, sci-fi, horror any genre, any time or place. All of time and space is open to Doctor Who. It allows creative freedom in story telling, its very nature is completely fluid and receptive to whatever story the writers wish to tell.
That's its great strength, its not wedded to that many rules as such. Its simply about a near immortal alien who can go anywhere and do anything. If the Doctor isn't restricted to any time or space or any kind of story, why place the limitation of a fixed gender on the Doctor as well. If the Doctor can be old, young, blonde, dark haired, English, Scottish, the Doctor can be anything. He can be anything. She can be anything. Could be black, ginger, even... American.
One thing is always a given certainty with Doctor Who and that's change. The Doctor changes and regenerates, the companions move on or die, but it always changes and carries on. Eventually a different actor will play the Doctor, will he be young, old, Scottish, Irish, Welsh, black, white. It depends what way they want to go, what stories the change would open up and allow them to tell.
As we can currently see an older Doctor has once again changed the nature of how his adventures and interactions play out. A female Doctor would open up more avenues, it would be interesting, which is surely the point, to tell interesting, entertaining and engaging stories.
The show evolves to fit and make use of these changing elements. Indeed, the very reason he regenerates isn't because they just decided to do it, nature dictated it. Hartnell got sick and couldn't carry on, but the show had to. So they thought their way out of the problem and established a brilliant plot device, which allows the show to continue forever.
As Doctor Who can tell and be any kind of story, it is often able to go places and show things which might not be currently acceptable culturally speaking. Indeed the very nature of its creation broke barriers, both on the screen and behind it, produced by a woman, directed by a gay Asian man.
As we have grown and changed culturally as a nation, its important that our popular culture both reflects and even at times leads us. Doctor Who has done this, if it were to have a female Doctor or even a gay Doctor, it clearly wouldn't be out of character, it would be acceptable, it would simply be a reflection of the society we live in. It wouldn't at all be a bad thing.
The simple truth is it boils down to how its written. If its written well and cast well, there is no reason the Doctor cant be a woman or any other race, the Doctor can be anything. As long as the Doctor is the Doctor, it will still be Doctor Who. Those saying they wouldn't watch it they would and if they learned a few lessons of acceptance along the way, that wouldn't be a bad thing.
A further truth is the target audience, the children, they aren't growing up to be in the world we knew, their world is going to be much more open to gender reversals and fluidity than ours was. They deserve to and should have that reflected in popular culture, so they learn to live it and see it as normal. As Doctor Who taught us about a world of differences when it came to race, now it can and should teach them about gender.
Really the only reason to object is because it would be new and it would be different. Not what I grew up with. Which is fair enough, I don't like the idea for that reason and trying to argue against it has made me realise it.
At the moment the only real barrier I can see to not doing it is the writing. Moffat has gone off a cliff, if he is in charge, it will probably be dreadful.