Answer

An entity that exists necessarily and then commits suicide and ceases to exist is like a square circle. So the question of whether He can commit suicide is comparable to asking whether He can make a square circle.
It reminds me of Puss in Boots, who comes to the palace of the terrible sorcerer, who immediately turns into a lion and threatens to devour him. The cat stops him and asks whether he also knows how to turn into a mouse, and when he answers yes, the cat devours him (and ever since, they all live happily ever after). Can the terrible sorcerer really be devoured after turning into a mouse? Or alternatively, can a terrible sorcerer really turn into a mouse in such a way that any cat would devour him? Can God turn into a human being and then I shoot Him in the head? Apparently not. And if, when I shoot Him in the head, He does not die, then He did not really become a human being. Either way, this is a self-contradictory task, so obviously there is no point discussing whether God can perform it. His committing suicide is exactly the same thing. In other words, it is not that He cannot commit suicide, but that the concept of suicide is not defined with respect to Him (just as it is not defined with respect to a stone or water, or with respect to kindness, or the trait of anger). Therefore your question cannot really be discussed.