How would he know if it is an illegal order? If it violates the Geneva Conventions, International treaties, etc. Just like a cop knows when his boss tells him to do something illegal.

His answer was right, but should have been qualified. 

The US Admiral isn't in charge of the entire nuclear network. There are enough nuclear weapons to destroy the world a few times over. So a fifth of US nuclear weapons would be enough (especially if the target is one country only).