There are many more deaths by drowning in backyard pools in places with more... wait for it... backyard pools.

People determined to kill themselves, as opposed to cry for help, will choose the most lethal method available, and the US has a much lower suicide rate than most countries with more restrictive gun laws. So gun ownership lacks any correlation, much less causation, to suicide in any actually meaningful way.

And, of course, most gun suicides (as in something like 90+%) are middle-aged, predominently male, gun owners so no realistic gun restriction can rationally have any effect. They already own the guns, and can pass background checks to buy one if not. Most serious suicides plan their deaths, so waiting periods are useless (they'll wait or substitute). As adults they have access to even a "safe-storage law" gun and ammunition. Only one shot is needed so any gun can work, even a flintlock musket, so semi-auto, pistol, mag capacity, etc bans can't help.