This is just a variation of the famous "Can G-d create a rock so heavy that He can't pick up" question. The answer: Yes, He could create such a rock, and He could pick it up too. Sounds illogical? Well, G-d was the one who created the rules of logic. G-d is not just unlimited, because that would be a restriction too - that He can't be limited. There is a famous quote from a book called "Avodas Hakodesh" that obviously G-d is both limited and unlimited or else there would be something lacking in Him. It's not something we can fathom, because the human mind cannot comprehend G-d.

I believe that this question is based on a premise that illustrates a shallow understanding of what G-d is. The question assumes that the rules of logic are somehow inherently there, and G-d is bound by them. If that would be the case, if we would say that G-d is unlimited then He would be bound to being unlimited, and could not also be limited - because logically something which is unlimited cannot also be limited. Another form that this question often takes is the famous "Can G-d create a rock so heavy that He can't pick up".

In fact, not only did G-d create the world, but he created logic too. He is therefore not bound by the constraint of logic. Yes, logically it is not possible to pick up something that is too heavy to pick up - but G-d was the one who made the rock, the concept of heaviness and the rule that something that is too heavy cannot be lifted. So the answer is yes, He could create such a rock, and He could pick it up too. Another example: G-d created a world in which one plus one equals two - He could just as possibly have made a world were one plus one is three.

The is famously expressed in a Kabbalistic work called "Avodas Hakodesh" (Section I, ch. 8):

אור אין סוף הוא שלימותא דכולא, כשם שיש לו כח בבלתי בעל גבול כך יש לו כח בגבול, שאם תאמר יש לו כח בבלתי בעל גבול ואין לו כח בגבול הרי אתה מחסר שלימותו ואין סוף הוא שלימותא דכולא - Just as the Infinite One possesses power with an infinite dimension, so too does He possess power with a finite dimension. For if you were to say that He possesses infinite power but not finite power, you would detract from His perfection [for there would be an area, the realm of finitude, that would remain outside the context of G-dliness -- which is impossible], for G-d is the ultimate perfection.

In other words, since G-d is the ultimate perfection, there is no advantage that He lacks. In logical terms, two opposite advantages can be mutually exclusive (How can something be limited and unlimited at the same time), but for G-d that is not a constraint.

So to answer your question: Can G-d destroy Himself? G-d possesses any advantage that there might be in being able to self-destroy, but at the same time this does not contradict with His ability to exist forever.