Short answer -- no
First, we suicides by guns are decreasing --
Okay, but what do gun ownership rate look like?
Thee ATF states the number of guns has gone up (quite significantly, it seems) even though the number of gun owning households has decreased --

:large
The first thing this tells us (assuming people are being honest with pollsters at a time when gun owners are being vilified in the media and culture) is: The number of gun ownership is being concentrated among fewer households.
That would seem to imply that increasing the more guns a household owns lowers the suicide rate.
But what about the overall non-gun suicide rate?
For the most part suicide rates are declining regardless of the method. There are exceptions, such as, hangings among women seem to be on the uptick as a method even though suicide rates overall are in decline.
In other words, suicide rates are not tied to gun ownership.