You can do that by comparing the rates in various states against the rate of firearm ownership. There does seem to be a positive correlation between firearm ownership and suicide in the United States, although other correlations are equally present and might be causative as well. Rural areas tend to to have high suicide rates[0], and tend to both be more poor and have more firearms than urban areas per capita.

Globally it's more complicated, as there are plenty of countries with fewer firearms than the US and more suicides, as well as countries with fewer firearms and fewer suicides. Material conditions (poverty, etc.), prevailing attitudes around substance use, hard to describe social well being factors, and stigma/quality of mental health care seem to be factors.

For my part, I think that easy access to a means of suicide, such as firearms, is one part of the equation, but not the dominant part. The dominant part is the prevailing experience in the population as a whole, as a population that feels like it's lost hope is more likely to attempt suicide on average than one that has not. But it does seem like wide spread misery and access to an easy and quick means of suicide is a very bad combination.

0 - https://www.cdc.gov/media/releases/2017/p1005-rural-suicide-...

Doing the comparison between South Korea and USA, restricting firearm access look to be increasing suicide rate. Of course this is not true, just an extreme outlier, the point is that cultural differences have a much stronger impact than access to a particular method.