I wouldn't really expect to see such a correlation because there are certainly other factors that have a significantly greater effect on suicide rates and there

I wouldn't really expect to see such a correlation because there are certainly other factors that have a significantly greater effect on suicide rates and there aren't that many independent countries (so you can't accurately measure from such a small "sample" of "possible" countries) and gun ownership isn't independent of other factors that are known to influence suicide rates: for example, the suicide rate tends to fall (I've been told) during armed conflicts while there is probably a positive correlation between gun ownership and armed conflicts. Probably you could reduce the suicide rate in a US State by (further) restricting gun ownership though it's anyone's guess by how much and it probably wouldn't make a huge difference, I would guess.