What irks me about this concept is the assumption that in order to be a "martial art," shooting must be done with robes, belts and Japanese or Chinese names.
But I think that's what the OP meant. If not, why the question? Anyway, once you've got a gun (and a reason to use it), I think all the other stuff becomes largely valueless. The gun so much more effective than anything else that punches, kicks, holds, throws, knives, sticks, and whatnot are reduced to near zero.
Anything other than a gun is only useful if you're in a situation in which you can't use deadly force. This pretty much relegates knives, sticks, and whatnot to curio status for self-defense.
There are many people who post on this BB who swear up and down that whipping out your gun and shooting unarmed people is just fine and that the police will give you a medal for that. I suspect these people are wrong for a variety of reasons. It's these situations in which the gun becomes useless at best and more likely a liability. That's when you need your hand-to-hand skills.
If the situation calls for the use of deadly force, then the gun is king, pope, and emperor. If not, then it's an embarrassing lump of steel on your belt and you'd better keep it out of the action while you try to kick, punch, or use your vibrating palm. Is there a "martial art" in the Bruce Lee sense that combines the two? I don't think so.