One good Star Trek point is that Kirk is SO in command that, despite the computer demanding 2 more officers to agree, the order is carried out with not a speck of hesitation. The other officers don't even consider any other course of action, as the 3-password activation procedure suggest they do. Kirk just issues his orders and keeps staring at the black/white dude (or was the white/black one? I forget).
Knowing him for 2 seasons at this pont, the ship's computer could as well save the trouble of asking the confirmation.