No, it wouldn't; it's equivalent to proving that defense must be run by the state. If you did some homework, you'd find out that the arguments for state-run defense are very thoroughly expounded out there. I would expect you to counter with that argument, or at least demonstrate some familiarity with it.
The going argument is that of Samuelson, who argues that defense is a "public good." A "public good" is defined to be something that is (1) non-excludable, and (2) non-rivalrous. The former means that you can't prevent free-loaders from being defended. The latter means that my benefiting from defense in no way diminishes your benefit from defense--or, crudely, "defense" is not a "scarcity good".
The rationale behind that argument is two-fold. First, non-excludability implies that free-loaders can refuse to pay for defense, but the defenders are forced to defend them anyway. Thus, people will not pay enough to bear the cost of defense (unless the government forces them to do so). Second, non-rivalry means that it is somehow "inefficient" to allow people to be undefended, since defending them doesn't diminish anyone else's defense. In effect, it's "inefficient," or "wasteful," or "harmful to society," if non-rivalrous goods aren't distributed to everyone who wants them. These arguments are considered so water-tight, that "national defense" is generally cited as the "purest" example of a public good.
The argument is pure buncombe.
First, excludability. Defense is not "perfectly" excludable, it's true. If my defense contractor ends the war by conquering the Mexican invaders, non-subscribers benefit from that. But it is certainly possible to exclude people. For example, a defense contractor might provide an air-raid shelter; non-subscribers can be refused entry. If in some town nobody subscribes, then my defense contractor won't defend that town, unless it happens to be in its subscribers' interest for strategic reasons. The contractor will conduct rescue missions, but only to rescue subscribers. And so on.
Second, non-rivalry. If a rescue squad comes for me, they might (or might not) rescue non-subscribers imprisoned with me; that's a non-rivalrous situation, if there are seats left in the rescue vehicle. More generally, however, defense is certainly rivalrous. Defense takes resources. Mobilizing here reduces defensive manpower over there. There are only so many guns and so much ammunition to go around.
What don't you try to prove that it can be? ... Defense of one's life, liberty, and personal property is not national defense.
The founders disagreed with you. Those minutemen at Lexington and Concord were fighting to defend their lives, liberty and property. If every citizen is armed and ready to fight, to the last man, in self defense, then the nation is well protected indeed. The invading Canadian army will run into a solid wall of armed citizenry. There's a reason Switzerland wasn't invaded in WWII, you know. The Nazis owned everything between Spain and Russia--except Switzerland.
Take Three guesses: Can you spot Switzerland?
Since service in the milita was compulsory...
I look forward to your proof of that statement. But before you wear yourself out, here's a free clue: Pennsylvania was just crawling with Quakers.