As for 'Earth Army', as adamant I am that there aren't or shouldn't be Starfleet Marines, I actually think there could be various planetary armies and militias.
I'm certainly in favour of Starfleet having access to a force of commandos/special ops et al, but I'm fine with them not being the "Starfleet Marines" (a light infantry/combined arms force centred operationally and psychologically around offensive warfare modelled on the USMC). Therefore if light infantry do exist (and Nor the Battle to the Strong bears this interpretation.
At least during the 2100s, Earth's military would appear to have been known as the United Earth Military (from an onscreen bio of Balthazar Edison), which would fit with the known pattern of the United Earth Diplomatic Corps, United Earth Space Probe Agency, United Earth Starfleet (possibly also known as the Starfleet Command of the UESPA), and the (United) Earth Cargo Service/Authority.
These however would be more like national guard type organisations, who help in cases of various natural disasters or defend the planet in the unlikely case of an invasion. In some places they might be the same thing as the local police or fire and rescue service. They are not deployed outside their homeplanets and would be run by the member governments, not the Federation. There is no 'Federation Army.'
I buy that de facto control of the PDFs would be the host planet governent or even sub-divisions thereof, as an active-duty reserve force primarily for planetary (or local system) policing, SAR, disaster relief, and first-line 'rapid response' defense. However, I would suspect that training and equipment is maintained to Starfleet standards (though perhaps not 'top of the line') and that like the US National Guard, British Army Reserve (previously the Territorials) and similar units in other countries, it is possible to 'detach' personnel to serve with Starfleet on a unit-by-unit basis within the same chain-of-command (for example, Wayeh Surt and Cenn Desca's role as Bajoran Liaison Officer to Deep Space Nine) and potentially even large formations or commands as needed during times of war (this is the only the source of infantry or armor divisions and above that I consider credible in Federation service).