No, not literally. (Although tell me that wouldn't be awesome. Perhaps the Pertwee Doctor? He could probably hold his own against Batman, he knows Venusian Aikido.) No, what I was referring to is that these are probably the two most prominent fictional characters who make a point of disliking guns. Other characters might not use them or carry them, but Batman and the Doctor are almost alone in very vocally disdaining them. (The only other that immediately comes to mind is MacGyver, who was of course created by a prominent Doctor Who writer, Terry Nation.)
Interestingly, neither one of them had their anti-firearm stance as part of their original character bible. Batman, of course, started out as a pistol-packing pulp hero, who notoriously gunned down his early opponents before Bob Kane and Bill Finger figured out that a) there's more money in being kid-friendly, and b) it's easier to write a continuing series if you don't keep killing off your best villains. The Doctor didn't start out with any particular opinion on guns one way or the other; he was an elderly inventor, an eccentric scientist with a brisk trade in improvised gizmos. It was only later, as he was put into a military setting, that his decision not to carry a gun became a character point instead of an afterthought. (Hmm, improvised gizmos, no guns...wonder where Terry Nation got the idea for MacGyver, exactly?)
So we have two characters who have a very similar aspect to their personality...except that they don't, do they? After all, the Doctor disdains beating people up pretty much in general. (Venusian Aikido notwithstanding.) He carries a screwdriver in his pockets--a tool, not a weapon. Batman, on the other hand, leaves the gun behind to carry weapons that he can apply with better precision and non-lethal force. He's more than happy to punch, kick, thump, and batter enemies into submission. He packs a Batarang, not a Bat-Screwdriver. (Except the Adam West Batman, who naturally packed both.)
So with that in mind, you'd think that it would be more shocking to see the Doctor pick up a gun, right? After all, he's the self-proclaimed "man of peace", the one who always gives his enemies the chance to surrender, the one who doesn't solve his problems with his fists. One would imagine that to see him pick up a gun and use it to solve his problems, well...that'd be a sign of a crisis so grave that it shakes his moral foundations to their very core, even more so than Batman (who, as mentioned, started off his career as a card-carrying NRA superhero.)
Except that both heroes have done exactly that, lately. First in 'The End of Time', then just this last week in Doctor Who (well, last week in Britain...next week in America...um, spoilers, okay?) the Doctor used a gun; and last year, Batman had a dramatic face-off against Darkseid that ended with him shooting the god in cold blood. Which was more out of character? Batman's shot, by far. The question is, why?
The answer can be found in their respective moral codes. It's actually a pretty interesting study in contrasts--Batman's moral code is very inflexible, the Doctor's...well, let's just say it's less so. Once Batman makes a decision (no killing, no guns) he holds himself to it with steely determination. The Doctor? He's against killing as a general rule, and he's always trying to find a non-violent solution...but when he comes up against an enemy that can only be stopped with lethal force, he's willing to use it. He's against guns, but not for the same reason as Batman--Bruce Wayne refuses to use guns because they're the cowardly instrument of death that wiped out two of the finest people he ever knew.
The Doctor doesn't carry a gun because he feels like they limit the mind. "If all you have is a hammer, then everything looks like a nail," as they say. If all you have is a gun, then you'll see every problem in terms of "What needs to be shot?" And when you come up against as many bullet-proof enemies as the Doctor, it can be downright suicidal if your first instinct is to reach for your gun. He doesn't carry one because it makes him tactically flexible, not because he hates the things. (Although he's not fond of them, either. But there's a difference between disliking guns and having a visceral hatred of them.)
So when the Doctor reaches for a gun, it means that the situation genuinely requires a gun. When Batman reaches for a gun, he's overcoming a lifetime of loathing and a traumatic association between the weapon and the man who used it to shatter his life forever. And that's why it will always be more dramatic to have Batman use a gun than the Doctor. Even though their philosophies seem alike on the surface, they really couldn't be more different. And despite the fact that Batman seems like the more violent character, it's really the Doctor who's willing to be more ruthless with his enemies.
(But tell me a team-up between them wouldn't be awesome. Batman and the Doctor fighting Ra's Al-Ghul and the Master? Oh, yeah.)