In reality, there is the two man rule in place if the US is not under attack. The Sec of Defense, or someone on the list if he is not available, must approve the launch as well. Like Jack Ryan in the novel, if he feels the President has lost his mind, he can say so, telling NORAD that the order is not valid and recommending they not accept any other order from him. Yeah it would cause a constitutional crisis, but better that than a nuclear war.
Unless the president has picked a complete wiener as SOD, he can't launch on his own. Beside, I would expect the leaders at NORAD, who actually understand what launching these things mean, to act on their own if we are not under attack and it appears the president is nutz.