[Content warning: suicide. Thanks to someone on Twitter I forget for alerting me to this question]
Among US states, there’s a clear relationship between gun ownership rates and suicide rates, but not between gun ownership rates and homicide rates:

You might conclude guns increase suicides but not homicides. Then you might predict that the gun-loving US would be an international outlier in suicides but not homicides. In fact, it’s the opposite:

Why should this be?
We’ve already discussed why US homicide rates are so high. But why isn’t the suicide rate elevated?
One possibility: suicide methods are fungible. If guns are easily available, you might use a gun; if not, you might overdose, hang yourself, or junp off a bridge. So getting rid of one suicide method or another doesn’t do much.
This sounds plausible, but it’s the opposite of scientific consensus on the subject. See for example Controlling Access To Suicide Means, which says that “restrictions of access to common means of suicide has lead to lower overall suicide rates, particularly regarding suicide by firearms in USA, detoxification of domestic and motor vehicle gas in England and other countries, toxic pesticides in rural areas, barriers at jumping sites and hanging…” This is particularly brought up in the context of US gun control – see eg Suicide, Guns, and Public Policy, which describes “strong empirical evidence that restriction of access to firearms reduces suicides”.
The state-level data from above support this view – taking guns away from a state does decrease its suicide rate. And then there’s this graph, from Armed With Reason:

…which shows that adding more guns to a state does not decrease its nonfirearm suicide rate.
But if suicide methods aren’t fungible, then why doesn’t the US have higher suicide rates? Here’s another way of asking this question:

The US has fewer nongun suicides than anywhere else. The seemingly obvious explanation is that guns are so common that everyone who wants to commit suicide is using guns, decreasing the non-gun rate. But that contradicts all the nonfungibility evidence above. So the other possibility is that the US ought to have an very low suicide rate, and it’s just all our guns that are bringing us back up to average.
Of all US states, Massachusetts, New Jersey, and Hawaii have the fewest guns. Unsurprisingly, suicides in these states are less likely than average to be committed with firearms. In MA, the rate is 22%; in NJ 24%; in HI, 20%. Their suicide rates are 8.8, 7.2, and 12.1, respectively. Hawaii has an unusual ethnic composition – 40% Asian and 20% Native Hawaiian, both groups with high suicide rates (see eg the suicide rate for Japan above). So it might be worth taking Massachusetts and New Jersey as examples to look at in more detail.
Either state, if it were independent, would be among the lowest-suicide-rate developed nations. And both still have more guns than our comparison countries. If we did a really simple linear extrapolation from New Jersey-level gun control to imagine a state where firearms were as restricted as in Britain, we would expect it to have a suicide rate of around 5 or 6 – which is around the current level of non-gun US suicides. This is much lower than any of the large comparison countries in the graph above, but there are two developed countries currently around this level – Italy and Israel. I think it makes sense to suppose that the US might have a low Italy/Israel-style base rate of suicides.
For one thing, it’s unusually religious for a developed country. Religion is one of the strongest protective factors against suicide. This also seems like a good explanation for Italy and Israel.
For another, it’s culturally similar to Britain, which also has a low suicide rate somewhere in the 7s. Other British colonies don’t seem to have kept this effect – Australia and Canada are both higher – but maybe the US did.
And for another, it’s unusually ethnically diverse. Blacks and Hispanics have only about half the suicide rate of whites; which means you would expect the US to be less suicidal than Europe. I previously believed this was because whites had more guns, but this doesn’t seem to be true: Riddell et al find that whites have higher non-firearm suicide rates too. So this could be an additional factor driving US rates down.
(another conclusion from the graphs above: US whites – who have most of the guns – do have an anomalously high suicide rate compared to other countries)
A confounding factor – the US has lots of different cultures, and Massachusetts and New Jersey represent only one of them. But if anything I would expect Southern and Midwestern culture, which are more religious, to have a lower base suicide rate; the South also has a lower percent white, another reason to expect their rate to be lower. And there is no evidence of these states having a higher non-firearm suicide rate, which we might expect if they were unusually suicidal.
So I think the simplest explanation is true. A gun-free US would have one of the lowest suicide rates in the developed world, maybe 5 or 6 people per hundred thousand. The US’ average-seeming suicide rate is an artifact caused by combining the low base with the distorting effects of high gun availability. The lack of a relative suicide crisis in the US doesn’t indicate that easy firearm access isn’t causing thousands of preventable suicides per year.
This is maybe not the most pressing question we’re facing right now, but I take it as a warning against gotcha-style debating. A simple bar graph comparing national suicide and homicide rates would be a compelling, elegant, and easily-digested argument that guns increased homicides but not suicides. It would also be totally wrong.
[EDIT: Commenters point out this paper by Alex Tabarrok on how there is some, but less-than-perfect, substitutability of suicide methods.]