This is not how it works, the president has the sole authority to decide to launch a nuclear attack. Quoting from Wikipedia.
> This verification process [involving the Secretary of Defense] deals solely with verifying that the order came from the actual President. The Secretary of Defense has no veto power and must comply with the president's order.
And from another article:
> If the Secretary of Defense does not concur, then the President may in his sole discretion fire the Secretary. The Secretary of Defense has legal authority to approve the order, but cannot veto it.
That quote is paraphrasing e.g. this source in the New York Times:
> “There’s no veto once the president has ordered a strike,” said Franklin C. Miller, a nuclear specialist who held White House and Defense Department posts for 31 years before leaving government service in 2005. “The president and only the president has the authority to order the use of nuclear weapons.”
Furthermore, the reference to the "button" (nuclear football) and the Secretary of Defense being involved is only a reference to the protocol around that specific launch system, but there's other methods of launching nuclear weapons available to the president, and which operate at his sole discretion.
Here's what ex-Secretary of Defense Bill Perry said about it:
> “What is clear, is that the secretary of Defense does not have veto power on it. This is a decision of the president’s,” Perry told Politico’s “Off Message” podcast.
> “The order can go directly from the president to the Strategic Air Command. The Defense secretary is not necessarily in that loop,” Perry said in the interview.
There’s also no obligation for the military to obey an illegal order.
The verification step is just that they have to acknowledge that the person ordering the launch is indeed the president. They get no say in whether it happens.
> There’s also no obligation for the
> military to obey an illegal order.
Once the launch order goes submarine crews or crews in various missile silos aren't going to be given the chance to check what Trump has been saying on Twitter, and they have no independent way to verify that North Korean / Russian / Chinese ICBMs aren't already in-flight heading towards every major city in the US.
OTOH, an illegal attempt by the President to order a nuclear first strike could conceivably be stopped before the order left the National Command Authority, and the apparent violations of policy would be legally defensible after the fact.
If the President orders a blatantly illegal, unjustified nuclear first strike, the outcome will be millions of innocent deaths and the probable complete devastation of the United States. Refusing to verify the order, physically isolating the President from the ability to communicate down the chain of command, or even more drastic actions could be papered over and excused after the fact a lot more easily than “just following orders” that would lead to the largest and most pointless loss of life in human history.