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.