If it comes down to a real, open conflict (rather than the government quietly acting against the League), there are two possible outcomes:
The League (as a whole - no doubt some members would refuse.) surrenders to full US government control without a fight, or the army is wiped (albiet with heavy losses among the less powerful members of the League who don’t defect to the government, and among the US forces.*).
The League has Superman, Supergirl, Wonder Woman, J’Onn J’Onnz, and Green Lantern, (and possibly Captain Marvel, although I don’t know if he’s in the animated 'verse) the five (or six) of whom could take the US Armed forces on their own. While the government apparently (see Fearful Symmetry) has some Kryptonite (from Luthor, perhaps?)**, they couldn’t have it everywhere, so Superman and Supergirl could use hit and run tactics to avoid it. J’Onn’s weakness is easier to exploit, but a) I’m not sure he has it in JLU, and b) I have no idea if the government would know about it. The Government would have Galitea (Who Superman and/or Supergirl could concentrate on neutralising, because the government couldn’t bring Kryptonite to bear in her presence) and is likely to have Captain Atom on their side, but he’s be playing catch up given the number of powerhouses the League could field. I don’t think he could tell the gov J’Onn’s afraid of fire, at least.
Batman would be a good inside-man (and is unlikely to defect), except for the fact that Waller - and presumably others in the government - apparently knows his identity. We can expect Bruce Wayne to lose his fortune soon after the conflict started. Probably Ollie (who totally wouldn’t defect), too, since if they figured Bats out, they’ll have gotten Green Arrow, too.
The government has a stable of superhumans of their own - The Ultimen and Suicide Squad come to mind immediately, and there’s probably others - but on the whole, they’re not very controllable, and they’re nowhere near the power level of the JL’s big guns.
A lot of non-American governments, and likely the UN, would probably form up against the US government - in fear of the weapons the US government would be having to bring to bear against the League, and of the League-as-US-weapon. Whether this would include sending troops to aid the League, or simply censuring the USGov, I couldn’t say. The US government would likely have allies, as well, of course. Again, though, whether they send troops or simply vocal support is an open question. I think if one set of allies sends in troops the others likely will, as well.
So…ultimately, the fight goes to the League, but it won’t be a happy ending. And whoever wins will have lost a lot of the people’s support that they had before.
One possible mitigating circumstance is how the Guardians of Oa interpret this.
If they see it as internal wrangling on Earth, they might leave it to John to sort out, since it’s his beat.
If, however, they saw this as a potential threat to the universe at large, they might send some more GLs (probably couldn’t keep Kyle, Katma Tui, or Kilowog from that group) to aid the League.
OTOH, they might think the government is in the right, and recall John, and send in (a) GL(s) to aid the government. This is a very unlikely possibility, IMO.
- Probable non-defectors***: Batman, Green Arrow, J’Onn J’Onnz, The Question, most of the non-American heroes (Fire, Ice, Rocket Red, Starman, Dr Light, Aztek, Booster Gold, Wonder Woman…), Supergirl, Superman, GL. Probable defectors***: Hawk, Dove, Captain Atom.
** This is assuming the project that created Tea was government-funded. I’m not remembering off the top of my head whether it was mentioned to be in the episode or not, although it seems to be part of Cadmus (Dr Hamilton was involved in both, wasn’t he?), and given Ultimatum and Dark Heart, it seems likely.
*** Based on character nationality, stated attitude towards authority, statements made in Dark Heart, or some combination thereof. (And just because he’s loyal to Hawk in Dove’s case.)