Examples:
"As surprising as it might sound, neither higher gun ownership rates in a state nor varied attitudes toward gun policy seem to correspond with more gun murders (or even with more gun robberies). This was still true in 2015 and now in 2016 as well. In the graph that follows, each dot is a state, plotted for gun homicide rate and gun ownership rate in 2016. The correlation here is too small to be statistically significant." (Washington Examiner; click on this link to see the graph in question.)
"Higher rates of gun ownership are not associated with higher rates of violent crime." (Heritage Foundation)
Response:
One would not expect higher rates of gun ownership to be associated with higher rates of crime overall, or even violent crime specifically, since guns are more lethal than other weapons (as discussed previously on this blog; see also this study). For this reason, it is not surprising that rates of lethal violence are exceptionally high in the U.S. than in other developed countries, whereas American rates of overall violence (ranging from assault to burglary to theft) are not exceptional compared to other developed countries. Similarly, a majority of homicides are committed with guns but a minority of violent crimes are committed with guns.
For these reasons, one would expect that gun ownership would be associated with higher overall homicide rates, since the presence of a gun increases the chance that a criminal encounter will result in the victim's death. And studies have found a positive relationship between gun ownership and the rates of serious (i.e. lethal) violence: for instance, in the 120 most populated counties in the United States (here), and across the entire country (here, here, here, and here). Moreover, if this relationship reflected more guns causing higher homicide rates, you would also expect to see a positive relationship between gun ownership and both overall and gun-related homicide rates, but not with non-gun-related homicide rates (or at least not as strongly). This is in fact what is observed (here, here, and here). At the cross-national level, findings are mixed, with some studies reporting that higher gun ownership does not increase homicide rates after other factors are controlled for. Interestingly, as this study notes, this may mean not that gun ownership has nothing to do with higher homicide rates, but instead that economic factors are also very important, a finding supported by other research. Other cross-national studies provide clearer support for a positive, monotonic relationship between gun ownership and homicide rates (e.g. of women, overall, overall again).
With respect to whether higher gun ownership causes higher crime rates, rather than higher homicide rates (a category that is notably broader than murder), the evidence is, again, somewhat unclear. Some studies suggest that no significant relationship exists between the availability of legal guns and violent crime rates, while other studies find a positive relationship between the two (at least with regard to firearm-related crimes). For an overview of studies on this subject, see here.
Long story short, more guns are associated with higher homicide rates in most studies on the topic.