Access to firearms has certainly been shown to affect suicide rates, especially among men, given how surprisingly impulsive suicides are. But it's not sufficient, given that there are other countries with higher suicide rates and fewer firearms.
Also, while America has a lot of guns, there's also a lot of variance. The stats are a bit hard to compute, but there appear to be some states in the US that have fewer firearms per capita than some other countries, notably Switzerland, Canada, and Finland.
It's also worth pointing out that suicide by firearm is largely a male thing. In the United States women attempt suicide at 3x the rate that men do, but they tend to die because of that at a much lower rate (22.4 for men, 6.8 for women). The rate of completed female suicides is currently rising faster for women than men, worryingly.
It's complicated, basically.
And as always discussing suicide creates a contagion risk. If you're experiencing suicidal thoughts, please know that help is available at the national suicide prevention hotline, 800-273-8255.