I predict that he does resign. I also predict that Bill Clinton is nominated to replace him as UN Secretary General, and the world will enthusiastically endorse Clinton's elevation.
Biblical end times, here we come ...
There is no way on earth that the UN member nations will EVER agree to elect an American as Secretary General. Not even Bill Clinton, who managed to persuade people that he was "the first black president" and no doubt would tell the UN that he would be their first third-world Secretary General.
The nations of the world already think that the U.S. is too strong. To give them control of the UN administration as well would be simply inconceivable. As it is, U.S. citizens play a disproportionately small role in the paid U.N. bureaucracy.
As I've said before, it's similar to the rule that an American will never become Pope. The U.S. already controls an inordinate share of money, power, and influence in the Church. The Cardinals from other nations would never agree to let an American, no matter how qualified, take that office. If that is true, than it is much more true of the UN, where the organization is far more riddled with greed, corruption, and jealousy than the Church has ever been in its worst times.
No one from one the permanent UN member nations on the Security Council can serve as Secretary General.