That particular idea has been widely explored in the literature. E.g. Fajnzylber does it in Inequality and violent crime, finding a significant correlation of 0.54 between income inequality and log of homicide rate. This is pretty strong by social science standards. The correlation with other types of crime is much lower.

Curiously, If you restrict to Europe, the correlation is negative, but it is positive if you restrict to East and South Asia, which has Gini coefficients and murder rates comparable to European countries.

IMHO murder rates are incredibly gun-dependent, and I don't meant it as a gun-control argument, because politics is downstream from culture, so they are gun-culture dependent, not gun-law dependent. (Pro-gun culture with restrictive laws just means a huge black market, like drugs.)

Anecdotally, it is not easy to find black market guns in Eastern Europe. The supply of the ex-Yugo civil wars and drunk Soviet soldiers dried up, the international dealers and organized crime simply do not care about the minimal profits they could make on retail, they want it wholesale into conflict zones and whatnot. It is not a good black market retail business, unlike drugs, customers won't return every day or week. Retail black market could be based people owning 10-20 guns, private collectors, and occasionally sell one, there are a lot of people in the US who are like that but almost none in EE.

Things like not having a lot of game around to hunt play a role. But more likely, there are only two stable equilibria, everybody or nobody having guns, EE is tending towards nobody, the US has so many already that the only possible equilibrium state is everybody.