As Enterprise puts it, Starfleet was once shorthand for the United Earth Space Probe Agency Starfleet (of Star Fleet)...
Star Trek has always been a bit unclear as to whether (even in post-Federation/post-Enterprise times) Starfleet is a Federation "navy" or an Earth navy. In Star Trek 6 and Into Darkness, we saw a roomful of Starfleet admirals - and they were all human.
We know that there are all-Vulcan ships, and all Benzite ships...(and early on in TNG, the idea seemed to be that the "Klingons had joined the Federation" - and they had their own ships...and the flew both the Klingon and Federation symbol.) It seemed like the Vulcan and Benzite ships were Starfleet...but maybe the ships were managed by the individual species worlds - kinda like how the the 23rd century, the Enterprise was both a Starfleet *and* UNESPA ship.
In DS9 they mentioned how the Bajorian Militia would have to be "incorporated into Starfleet" if Bajor joined the Federation (and, btw, a "milita" is a military or quasi-military organization.) I wonder if the Bajorian branch of Starfleet would have been run from Bajor more-so than Earth?
In "Star Trek" '09 Pine said that "the Federation...in a humanitarian and peacekeeping armada (I think he really meant *Starfleet* was that - and I would have added "scientific exploration" to that.) Peacekeeping is a kinda military thing.
And in Star Trek 6 we had character with the rank of "major".
And I think that the MACOS were eventually absorbed into Starfleet...just look at their patch: http://images2.wikia.nocookie.net/__...o_insignia.png
^I'm betting that's where Starfleet got that elongated star-shape as part of their emblem/symbol...