That is a fascinating question. I found myself with a quick answer and then, as I began to write, realized the quick answer was not consistent with the name of God. Let me briefly share a thought...
In Hebrew, the word for name (shem) deals more with the character, reputation, and authority of the name bearer than it does what we call that person. So to "walk in the name of God" means to walk in a manner consistent with His character and within the scope of His authority. To profane God's name is to stand in opposition to His character, take from His reputation, or walk outside His will/authority. So when it comes to God's name, what "character" attributes describe Him? Well, the obvious;
love, joy, peace, patience, kindness, goodness, faithfulness, gentleness and self- control (Galatians 5:22-23). You might also consider Exodus 34:5-8. But what else do we know about God that describes Him, builds His reputation, adds to His character? He is LIGHT is He not? How about LIFE?
There is one more thing to consider. God has made certain promises. If He were to self-terminate, He would profane His own name not only for destroying true light and life, but also because it would mean He made promises He did not fulfill. So my answer is.... He might have the ability to, but it wouldn't and couldn't ever happen without Him profaning His name. And seeing His name is ETERNAL, He already knows it will never be profaned.