I don’t know how our intervention on England’s side squared with the Monroe Doctrine.
IIRC, the Monroe Doctrine only talks of the U.S. opposing further European colonization in its sphere of influence and not about existing, recognized colonies such as the Falklands.
The Monroe Doctrine addressed European interference with countries in the Western Hemisphere, it did not address existing European colonies. The UK was not attempting to invade Argentina and the people of the Falklands did not want any part of Argentina’s screwed up polity.
At the time the outcome of the Cold War was not at all clear, and the UK was our single most valuable ally.