I am sure you are looking for a sociopolitical reason, but as someone who lives with one foot on each side of the Atlantic I can tell you something, so to speak, at a micro level: people in Europe do not have as many guns as Americans for the same reason they do not have the same amount of obese people.
That is, the social stigma associated with it is very big, and deters guys who might here in the US fantasize about playing with guns from actually getting one. You can think of it in the same way many Americans think of cigarettes (and in this the roles are reversed).
Guns are in general considered gross, just like SUVs, baseball caps and white socks. And just like cigarettes in the US. If you are 20-something in Paris or Rome and your circle of girlfriends finds out you have a gun, you might as well erase their telephone numbers and e-mail contacts. Just like being a smoker severely damages your options when it comes to finding a date in New York City.
This is a very strong force in shaping social habits and opinions. If you know all the people around you are going to consider you a weirdo if you do something, you are very likely not even to consider that thing. Which is what most Europeans do. Owning a gun (apart from hunting, which also is something almost non-existant among people below 40) is something most people never think about. They don't refuse the idea. They never even consider it.