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.
I think if we weren’t going to support Argentina at the time (the military government being an ally against communism), we should have stayed out of it altogether. We damaged relations with Latin America for decades with our actions.