My hunch is that if there are fewer illegal guns, then you'll have less gun crime.
I don't think legal guns are a significant part of the gun problem, and its very rare to hear of someone legally carrying a handgun committing a crime (And if they do, its usually not a random crime).
The problem is, no one has yet figured out a good way to reduce the number of illegal guns . . . yes, there's some bleed over from the legal ones, but that's probably not the main source.
Frankly, my personal beliefs aside, I think that a serious gun ban in the US would be futile. People who are inclined to obey the lay would get rid of their guns . . . but they wouldn't be the people who comitted gun crimes anyway. The crminals would still have guns. A bunch of otherwise law abiding citizens who believe in gun rights would suddenly become criminals . . . and THE NUMBER OF GUNS WOULD BE EXACTLY THE SAME. They wouldn't go anywhere . . . . and more would migrate into the hands of criminals. I suspect that if you had fewer guns to start with, a gun ban might have some meaningful effect (see, e.g. Japan). But here, forget it.
So, even leaving aside American culture and the Second Amendment, I see no reason to make a bunch of law abiding citizens who like guns into criminals.