This is intuitive but wrong, in three ways.
(1) Domestically, the US Army does not enforce law or make sure the dollar is the "coin of the realm." That would violate Posse Comitatus act. US domestic law enforcement enforces laws, domestically - including legal tender laws. I agree you can apportion some of the strength of the US dollar to its legal system, which makes sense, as the currency is an emergent property of the state. However, shifting to BTC instead of USD will simply reallocate the expenditure not reduce or remove.
Counterfeiting is an issue broadly. A small one. Of course, the expense of a Bitcoin transaction (literally 60 days of power for an average US household and 1 iPad of e-waste) far, far, far outweighs the expenditure of the secret service whose job it is to act on counterfeiting. AML and KYC rules, and a counterfeit detection pen will prevent this fake money from turning into a digital representation.
The army isn't generally called in to stop counterfeiting because nation-states have basically zero incentive to counterfeit on a large scale. What good are a few fake paper bills, when the Treasury would just cut you off from the entire world financial system?
(2) How do you explain that countries that do not have armies at all, like Iceland, are able to issue their own fiat currency with value?
(3) If you cannot draw a direct line between switching the currency and reduction in military spending, does it really make sense to apportion value that way?