Maybe you got the relationship backwards. Maybe the countries that have lower crime gun crime rates just don't feel the need for gun laws.

One thing I've tried and failed to find on this subject is a breakdown by kind of guns. The vast majority of gun crime uses handguns, so I've been interested in finding data on handgun ownership rates and seeing if there's any relationship. But all the data I can find comparing gun-ownership rates to gun-homicide rates includes both handguns and long guns (rifles, shotguns), which have very different crime characteristics. For example Sweden actually has quite a few guns, but they are almost all rifles in the rural far north, not handguns in Stockholm or Malmö, which I suspect makes a big difference.

I'm not sure the data I'm looking for would actually show anything interesting either, but it's kind of odd that I can't find it at all.

Correlation does not equal causation.

Lack of correlation is evidence of lack of causation.