It’s actually likely an inverse relationship to this. Suicides are more likely to be successful where guns are prevalent compared to where they are hard to get. Meanwhile homicides are rare so no matter what tool you have access to you can get it done.
To be clear, I agree with that assessment. My point is that banning standard capacity magazines and semiautomatic rifles will have little effect on suicide rates, since they aren't a very good choice for committing suicide in the first place.
According your source in 2016 the US suicide rate was 13.7 per 100,000 people. The UK was 7.6, so about half.
But, France was 12.1 and Ireland 10.9, not much less than the US. I don't think we can conclude from this data that access to firearms accounts for all or even most of the difference.