Pretty sure the Monroe Doctrine says something to the effect that we don't want Europeans to subvert or try to re-establish a hold on their former colonies in the Americas.
Also pretty sure nothing in the doctrine says we have to support an act of aggression by a South American country against one of our European allies.
No it doesnt. What the Monroe Doctrine does is more a declaration of spheres of influence. At the same time it was a declaration that America would stay out of continental European affairs and their affairs on the other continents. While Britain was considered part of Europe the Doctrine was aimed more at the colonial powers of France, Spain, and Russia. Remember in 1823 we had no idea what the future America was going to look like but we did know we were going to expand west and south and we didn't want anyone stealing land in front of us. And we sure as heck didn't want to pay for it like we did in 1803 when we bought the LA. territories from France. This westward expansion, precluded by the Monroe Doctrine, was later termed "Manifest Destiny".
So the doctrine does not specifically say we have to attack anyone for anything. It was a warning and a statement of policy.
And even after fighting several wars with Britain we still had a love/hate relationship with them in the 1800's. Thats what happens when you share so much culturally at the beginning. Eventually this matured in the 20'th century becoming what is known today as "The special relationship".
I remember this Falklands war very well and remember that goofball Yank UN ambassador Jeane Kirkpatrick poisoning our relationship with England with her stupidity. The Yank public was overwhelmingly Pro-British and the Reagan administration eventually quit walking the fence. It must be said that The Cold War was in full swing and the Soviets/Cubans were looking for every opportunity to expand their influence in central and south Americas. The Argentine Junta was extremely anti-communist and were considered an ally in this regard, even if they used terror at home. It was the age old conundrum of Cold War reality. Do you back non-Democratic Govt.s that are anti-communist or do you sit back and allow a greater evil, communism, to take hold in the western hemisphere?
Part of what helped cause the Original Falklands war was that the Argentine Junta had convinced itself that America would back them in event of hostilities, or, at least not get involved. Partly the Reagan administration failed to get across to them that we wouldnt back them. The Argies had no idea of how the Yank public feels about the British. Today I think they finally understand the strength of those ties and since there is no cold war, and communism is no longer a threat, there isn't even that reason for America to not get involved. Partly what cause the war is the Brits had sent out mixed signals they would either not defend the islands, or couldnt defend them.
But it also must be said that even the Brits were surprised the Argies went for it. At the time it was pretty obvious they really had small hope of beating the Professional armed forces of Britain. To make matters even worse they attacked months before they had planned to and only had a small amount of exocet missiles in stock. The only Argie service that did well was there air force but even they took heavy casualties.
And If I remember right we imposed a arms embargo on Argentina during the Falklands war while speeding up arms deliveries to the Brits as well as providing electronic Intelligence to them. America provided out latest sidewinder AA mssiles and anti-radar missiles as well as gobs of jet fuel. The Argie junta had no means of replenishing their arms and even France refused to sell them more exocets.
Boy that has to be a low point when even the French wont sell you arms.