Is there anything that says The Doctor must be a man? I would love to see The Doctor come back as a ginger girl, and my friend tells me he thinks The Doctor cannot change gender. Is this true? If so where does it say that?
There's no reason that the Doctor would have to be a man upon regeneration, in-universe. Regeneration can cause various changes, including skin colour, outward species, and sex. The Corsair is/was a Time Lord/Lady who has changed gender from male to female. It's actually worth pointing out that in the charity comedic spoof of Doctor Who, Doctor Who and the Curse of the Fatal Death, the Doctor actually regenerates into a female at one point. The spoof, while produced by the BBC, is still technically outside canon, so as of this writing, there has not been an official female Doctor Who actor.
In "The Night of the Doctor" mini-episode, as the 8th Doctor is dying, he is offered to choose what characteristics he will regenerate into.
Since this states that the change is random, and since the Doctor has not become a woman (excluding Doctor-Donna) in all of his regenerations, it is unlikely that Time Lords can change gender by chance during a regeneration. However, this also confirms that a gender change remains possible, i.e. by harnessing the elevated Time Lord science on Karn.
Additionally, as seen in the Series 8 finale,
The Master appears to have regenerated into a female, "Missy", which shows that Time Lords can regenerate into other genders, although again, probably not naturally.
In Series 9, Episode 12 “Hell Bent”, we can see another Time Lord who regenerates from a male form into a female form. She mentions that her normal form is a woman and that she had been a man only once.
In this episode, The General (played by Ken Bones) regenerates into The Female General (played by T'nia Miller).
The regeneration scene is described in the script as follows:
All the technicians, shielding their eyes from a terrible golden glow in the centre of the room – a fountain of regeneration energy! Dashing through the door, Gastron – sees this.
GASTRON: (Into communicator) Med team to sector 52, Extraction Chamber Seven. Regeneration in progress.
The golden glow snaps off, a figure on the floor now struggling to sit up.
GASTRON (cont’d): Are you all right, sir. (Blinks at what he sees) Sorry, ma’am.
The General, sitting up, is now a rather younger woman. She’s a little groggy.
THE FEMALE GENERAL: Oh – back to normal, am I? Only time I’ve been a man, that last body. Dear Lord, how do you cope with all the ego?
This article seems to indicate not only is it possible to regenerate to a woman, but that it may actually happen.
Additionally some people have theorized that the events in The Angels Take Manhattan may lead to a regeneration to a female.
The answers here take care of the canon, indicating sex change is fair game, and that a Time Lord is not gender-bound.
Pyrodante's answer links to an article from the Daily Mail (Aug 2012) that suggests it's not inconceivable that we see a Doctor Her (or a Time Lady as they so eloquently put it).
My answer focuses on this article (Nov 2013), also from the Daily Mail, which suggests that such a thing is unlikely. The conveniently placed tl;dr at the top states:
Ex-Doctor Peter Davison insists the character should always be a man
'To have a female would be like having a female James Bond,' he says
However there was speculation that Georgia Moffett, daughter of Peter Moffett (stage name Davison) and wife of David Tennant could get the role, so the BBC has certainly considered that scenario, and has decided against it. At least for the time being.
The Doctor : She was my mancrush.
Bill : I'm sorry ?
The Doctor : Yeah, I think she was a man, back then. I'm pretty sure I was, too.
According to the BBC, the Doctor is genderfluid.
Personally, even though Moffat wrote in details that suggests he could regenerate into a woman, I think this has no basis in the logic of nature. I will explain.
There are two basic ways a species can reproduce, sexually or asexually. Since Time Lords appear to have two distinct sexes I can guess that naturally they reproduce sexually. (Any reproductions using technology are not important in this matter.) This would suggest then that they would retain this gender even through regeneration. Unless there was a need, by nature, to change the sex. There are species which change sex but it is always as demanded by nature or as a part of every member of that species lifespan, never at random.
This is why to me, it just doesn't make sense. If you used some kind of outside force to manipulate the regeneration into whatever you wanted, that's different. I understand Time Lords are special, But to me it's just never going to happen naturally unless you show me a clear need to the Time Lord species survival that some of them need to change gender.