It seems to me that the authors of this study have identified but not really controlled for the endogeneity problem that gun control regulations are both caused by and the cause of the gun prevalence. They report:
Access to lethal weapons is an important risk factor for suicide. Our study suggests that general barriers to firearm access created through state regulation can have a significant deterrent effect on male suicide rates in the United States. Permit requirements and bans on sales to minors were the most effective of the regulations analyzed.
My instinct is that decreasing access to guns will just lead to more non-gun suicides, which is what one study they cite actually found:
An evaluation of the 1996 National Firearms Agreement (NFA) in Australia documents a decline in firearm suicides after the implementation of the agreement (Klieve et al., 2009). However, these findings may be confounded with an overall decline in gun ownership that preceded the NFA. Additionally, there was some evidence of increased suicides by hanging.
However, I can imagine some people might deterred from a temporary impulse to kill themselves, and that the extra time that buys them might be enough to get help, change their mind, etc. Nevertheless, it is an interesting question, and I think some clever scientist out there should be able to think of an actual instrumental variable to get a better answer to this question.