I believe that taking a look at the issue at hand without looking at the legal ramifications would allow me to conclude that fewer guns would mean fewer suicides. Granted, I realize that people who are suicidal will find another method for suicide. I take into consideration that this question is about the fact that this would help prevent suicides. The folks on the other side of this issue argue that suicides will not be stopped. That is true. But one of the most prevalent types of suicides caused by guns is accidental suicide and that margin can be reduced almost completely if guns were completely banned. This is hypothetical of course since it would uproot the constitution in the U.S. and that is not likely to happen. As for those who are suicidal, tying guns to background checks would not only reduce access to the physical gun, but also potentially sound the alarm for people to get more involved to help unstable individuals.
If a person has his or her mind on committing suicide, they will find a way of doing so. A gun is not the only alternative to committing suicide. A person who wants to commit suicide will not only find a way , but look for different options as well.
The use of a gun--while it is an effective means of committing suicide--is not the only way of doing so. Reducing access to guns will not deter an individual who is adamant about ending their own life. They will find a way whether access to guns is reduced or not.
A person bent on suicide has a lot of methods available without needing to have a gun, and different people will prefer different methods anyway. Not having a gun available does prevent the sorts of situations where people can kill themselves in a moment of despair without having a while to think about it, but other than that I think people will kill themselves with any means available once they have decided to do it.