Robert DeSoto
Lieutenant Commander
In TOS Starfleet was the military. It was depicted as one talked about like one, its personnel behaved like military personnel. It was a military by every relevant definition of the word. Problem was there was no instance where anyone directly referred to Starfleet as the military. Which left wiggle room when the other series began denying Starfleet was a military, which stems from nothing but ignorance from the writing staffs of those shows about what a military actually is.
Sadly, there are more than a few lines of dialogue which claim Starfleet isn't military, which is why a lot of fans stick so passionately to that argument. Even Trek XI tries to dance around the issue by referring to Starfleet as a "humanitarian peacekeeping force." Even though that still sounds like a military to me, we're supposed to pretend that it doesn't.
Actually, there was. In the Wrath of Khan, Starfleet was called the military several times. Admiral Leyton's attempted coup d'etat was called a military dictatorship. There might be more, I don't know.
I agree with the OP that Starfleet is the Federation's military and all this exploration talk is Federation propaganda. Just like how the Defiant isn't a warship, its an "escort vessel". Even the Klingons called the Constitution class starship a battle cruiser. When the TNG crew lost its memory, Worf believed the Enterprise was a battleship based on its armament. All of Starfleet's "ships of exploration" are so well armed that anyone not subjected to Federation propaganda would recognize them as warships.
But to play devil's advocate, it is true that the primary mission of Starfleet has always been peaceful exploration and scientific research, like NASA. But in addition, Starfleet is also given the responsibility for protecting the Federation. So in that sense, it is not a military organization because although defending the Federation is part of Starfleet's mandate, it isn't the primary function.
The "peacekeeping armada" makes no sense however. Armada, yes, peacekeeping, no. Starfleet has never been involved in peacekeeping missions. They would probably think it violates the prime directive to deploy Starfleet security officers as a peacekeeping force.