Here is an article from Michael Shermer's Skeptic Magazine that gleans a result such as I described from a study comparing the degree of religious affliction to the level of "moral health" in a nation, as measured by crimes.
In turn the Skeptic article refers to
this study.
While I referred to violent crimes in a general way in the OP, the study above focuses at least in part on murder specifically. This prolly makes very good sense as the definition of murder is widely agreed upon, while some other types of crime might be considered violent by some but not by others, for example, a Catholic priest seducing a teenage child Some would call this rape and classify it with other violent crimes, others, while still despising the Holy Roman Child Rape Church might consider this crime not to be a violent crime, though highly despicable.
Generally folks agree about murder, except when the person performing the deed is wearing a uniform that arguably implies societal approval of the killing.
Despite a significant decline from a recent peak in the 1980s (Rosenfeld), the U.S. is the only prosperous democracy that retains high homicide rates, making it a strong outlier in this regard (Beeghley; Doyle, 2000). Similarly, theistic Portugal also has rates of homicides well above the secular developed democracy norm. Mass student murders in schools are rare, and have subsided somewhat since the 1990s, but the U.S. has experienced many more (National School Safety Center) than all the secular developed democracies combined.
At a minimum we can take away the knowledge that a higher percentage of religiously deluded citizens does not decrease violent crime, in fact it does appear to correlate with more violent crime.
The study goes on to find that belief in religious delusions shows a negative correlation with belief in the reality of science and evolution. Well, DUH!
The absence of exceptions to the negative correlation between absolute belief in a creator and acceptance of evolution, plus the lack of a significant religious revival in any developed democracy where evolution is popular, cast doubt on the thesis that societies can combine high rates of both religiosity and agreement with evolutionary science. Such an amalgamation may not be practical. By removing the need for a creator evolutionary science made belief optional. When deciding between supernatural and natural causes is a matter of opinion large numbers are likely to opt for the latter. Western nations are likely to return to the levels of popular religiosity common prior to the 1900s only in the improbable event that naturalistic evolution is scientifically overturned in favor of some form of creationist natural theology that scientifically verifies the existence of a creator. Conversely, evolution will probably not enjoy strong majority support in the U.S. until religiosity declines markedly.
This further supports my long held belief, based on over 6 decades of personal observation, that the blind acceptance of stupid ideas (theism) drives out the capacity for understanding what is real (evolution).
It really is Religion versus Reality, it is a real and important struggle, and history will find that the failure of the American people to reject idiotic bullshit was a major factor in the decline of the US Economy - by way of poor science educations causing poor job performance causing economic collapse. This process has already begun. As people in the grown-up nations of the world continue to adopt a reality-based lifestyle choice their understanding of what is real and ability to implement that knowledge in the form of technology will sail right past the glory days of the American Empire.
We will be left with high priced TV dinners and extremely banal reality TV. You get what you pay for.