It's a hard question to answer, since it never came it that. However, Nixon's Chief of Staff, H.R. Haldeman, did mention in his memoirs that Nixon's political strategy of "The Madman Theory" in attempt to feint the North Vietnamese:

I call it the Madman Theory, Bob. I want the North Vietnamese to believe I've reached the point where I might do anything to stop the war. We'll just slip the word to them that, "for God's sake, you know Nixon is obsessed about communism. We can't restrain him when he's angry—and he has his hand on the nuclear button" and Ho Chi Minh himself will be in Paris in two days begging for peace.

Haldeman, H. R. (1978). The Ends of Power. Times Books. p. 122

The plan was to let the idea of the "Mad Man" [Nixon] slip to the Soviets, indicating that he was unstable and could potentially do something drastic. While this a supposed political strategy by Nixon in dealing with the Soviet geopolitics, it's doubtful that he would ever initiate a nuclear war over an domestic snafu such as Watergate (as stressful and humiliating as the experience was to him). Most likely, the SoD Melvin Laird wanted to keep the ruse going for the Soviets' sake.

In the final days of the Nixon administration, Secretary of Defense James R. Schlesinger did tell the head of the Joint Chiefs that they should seek his approval before executing "any emergency order coming from the president." It's not clear that he could legally abort anything the President ordered — but I think Schlesinger was probably legitimately concerned about Nixon's mindset in his last days on the job. Apparently some of the Joint Chiefs thought that this was not about Nixon putting his finger on the button but in that he might attempt some kind of coup. Schlesinger did not clarify his position. Note that just because Schlesinger felt this way does not mean that Nixon was actually that unstable. But still.

Source: James Carroll, House of War: The Pentagon and the Disastrous Rise of American Power, 355-356.

Nixon went into a deep depression around his resignation. He was criticized much.