The answer seems to be that he cannot. Even the Omnipotent cannot do what is by definition impossible. What makes this difficult to see is the meaning of time and its relation to eternity. To live even for a day is to exist eternally. That is to
say, that day exists forever. It will never not have been. For it to cease to exist altogether would be for it never to have existed at all.
We speak of "ending the existence" of someone, but to annihilate his existence at time t + 3 is to annihilate it at time t, t + 1, t +2, and so on. A person's existence is the root from which their successive moments of life flow into reality. To destroy this "root" is to destroy everything they ever were, or did, or thought, or experienced, along with all the effects they ever had, both good and evil. It is to unpick the tapestry of which they are a part.
Perhaps, but there is still the problem that this would seem to contradict the stern words of Scripture. Even the mysterious prophecy about the death of Death (Rev. 20:12-15), would seem only to confirm the doctrine of an eternal burning in the "lake of fire".
"To the thirsty I will give from the spring of the water of life without payment. The one who conquers will have this heritage, and I will be his God and he will be my son. But as for the cowardly, the faithless, the detestable, as for murderers, the sexually immoral, sorcerers, idolaters, and all liars, their portion will be in the lake that burns with fire and sulfur, which is the second death” (Rev. 21:5-8).There are other ways in which God's mercy could be manifested. Some believe that the "time" in which the sinner experiences his punishment contracts to an infinitesimal moment. Just as the moment of death expands infinitely for those who live with God, it contracts infinitely for those who reject him. These and other views are discussed in more detail in The Radiance of Being.
But perhaps we do not need to worry too much about the details. "He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning, nor crying, nor pain anymore, for the former things have passed away" (Rev. 21:4). Our God is more merciful than us, and any possibility of kindness will have occurred to him long before it did to any of his creatures.