Well, the Falkland War was also a diplomatic issue for the United States. It didn't want to apply the Monroe Doctrine because of that, but it certainly didn't agree with Margaret Thatcher of the UK in this instance. The conflict actually led to some friction between Reagan and Thatcher, which was fairly rare apparently.

The U.S did everything it could to find a solution to the conflict, with Secretary of State Alexander Haig shuttling back and forth doing his best to try to solve it before the British entered the warzone. The U.S had no real good options in this scenario, however. It could not force Britain out of the Western Hemisphere, and it was concerned over the prospect of escalation. It could not find a solution to either side's wounded pride, and it was scared that ties with both Latin America and Britain would be hurt by the conflict between them. This was a lose-lose for the United States.

The United States took an initial policy of "even-handedness", arguing that both Britain and Argentina were allies and that they should work this out peacefully. However, neither party was very willing to listen. They believed that the U.S stake in the conflict made it someone they couldn't trust, and its hesitation to pick one side over the other made both more suspicious of it. Argentina figured that if the U.S believed it was an ally to both and wanted peace so badly, it wouldn't intervene on the side of the British...just as it hadn't when the Argentinians first invaded the Falklands. And sending Haig didn't seem to work very well, after all...he wasn't a great diplomat on this issue, didn't seem to know it well or come up with solutions quickly at all, and his "shuttle diplomacy" was a far cry from Kissinger's. And to add to this whole mess, Haig was disliked by Thatcher and by Argentina's leaders, on a personal level. He just wasn't the man for the job, and he didn't do much to help mediate in the end.

When peace overtures ended up rejected, the U.S decided to side with Britain. It viewed the British alliance as more important than the Argentinian one (meant to be against Communism), and decided to sanction Argentina while providing material support to the British. Argentina's calculation that it had leverage over the U.S, because of its role in the war against communism and in the protection of shipping lanes in the South Atlantic, was a miscalculation.

The British were upset when the U.S Ambassador to the United Nations, Jeane Kirkpatrick, made an even-handed statement justifying the use of force to pursue territorial claims based on historical grounds, which could be perceived as applying to the UK's right to respond or the Argentinian invasion alike. When questioned on it, Kirkpatrick explained her reasoning. She pointed out that the U.S had a longstanding commitment to the United Kingdom, but that it shared a common heritage with Latin America, which supported (besides Chile) Argentina in the dispute. So the U.S was caught between, as I said earlier, the fight against communism in the Western Hemisphere and the attempt to lead NATO against communism in Europe, which taking a side would not help. After all, Argentina had trade ties with the Soviet Union and refused to enact the embargo the U.S placed after the Soviet invasion of Afghanistan, which worried the U.S into thinking that the Soviets could cause mischief through Latin America. But it couldn't allow Britain to fail, either. That would be a huge blow to its pride, it would blame the United States for not exerting leadership and helping it, and it would demoralize the NATO alliance through Britain's anger.

By the time Haig negotiated, both sides were fairly deeply committed. Argentina could not withdraw or General Galtieri would lose all popular support and likely be deposed, and Thatcher had already sent a sizable Task Force that couldn't simply withdraw now that its pride was wounded. Argentina believed, with some good reason, that the UK was not serious about negotiations brokered by Haig. They seemed happy to allow negotiations to chug on while their ships chugged on, to paraphrase the words of one British War Cabinet member. Still, Britain had moved some on its proposals for a peaceful solution.

By April 26th, the stage was all but set. The British stopped all aircraft and ships going through the Falkland zone by April 28, and Haig gave his proposal to Argentina the day before. The Argentinians saw the proposal (which they had no fore-knowledge of) as coming on the tail of the incoming British threat, since it was handed to them on April 27. They wanted time to consider, not unreasonably, seeing the situation as prejudged and unfair. But they rejected the proposals, and so on April 30 Haig declared that the U.S would see the use of force to resolve disputes as unlawful, effectively siding with the British on this one. At that point, the U.S was committed to ensuring the British won: a loss would be devastating. The fact that mediation failed didn't get helped by the fact that Reagan was virtually in disbelief. Who would fight over such a ridiculous patch of island, he figured? Well, he underestimated the pride factor, it seems.

So why not invoke Monroe?

Well, the answer becomes clear above. The U.S didn't know who to value more, between the feeling that one had to keep united against the "communist threat" of the Soviet Union, and the feeling that Britain was a longstanding ally who needed this help more. Eventually the U.S judged British support to be more valuable, seeing it as necessary to maintain the European bloc's morale and help the allies.

But it did lead to a lot of friction between Thatcher and Reagan, as I mentioned. Reagan was floating peace initiatives to Thatcher, and it didn't get anywhere easily. On May 31 Reagan proposed one to Thatcher, as he was concerned over Latin American opinion of the United States. He made another attempt on his visit to Europe June 2-4. Reagan figured that now that Thatcher had the upper hand and British pride, she could negotiate. Thatcher replied:

that we could not contemplate a cease-fire without Argentine withdrawal. Having lost ships and lives because for seven weeks the Argentines had refused to negotiate, we would not consider handing the Islands over to a third party

That didn't make Reagan happy, but his warm relationship with Thatcher helped soothe it. Even so, the fact that over the May 31 phone call she rejected his pleas three times, and more strongly each time, was not helpful to the relations. This NY Times article on released documents from the National Archives talks about the friction that resulted from the dispute. But like I said, a warm relationship helped smooth it over. Which, coincidentally, may be another reason why the U.S didn't invoke the Monroe Doctrine. It got along so well with Thatcher, what use would invoking the Doctrine really be?

As was already brought up by /u/jgrey12, the idea that the Monroe Doctrine only applies to new colonization, and the Falkland conflict was over already held land, could also have influenced the way it played out. But it's also important to note that the Argentinians were disputing the Falklands being held by the British in the first place, so it could still apply. More importantly, however, the Monroe Doctrine was used to justify attempts to keep Soviet influence out of the Western Hemisphere, and this definitely played into the conflict: as I said, the U.S was reluctant to go against Argentina because it feared Soviet influence, after all. But it chose not to invoke the Monroe Doctrine in siding with Britain, because it believed that the British were more important than possible Argentinian turns to communism, which could really have led to an invocation of the Monroe Doctrine since that would be expansion of influence unlike the British case which appeared to be more preserving-based.

Actually an interesting corollary can be found in the case of Grover Cleveland's handling of the Venezuelan Crisis of 1895. Venezuela and Britain had a dispute over who had the rightful claim to Guayana Esequiba, and the U.S sided with Venezuela in this case. It also had the Monroe Doctrine invoked, and the British effectively submitted to it in this case. So in cases where control is disputed, as with the Falkland War, it can be down to more than just whether or not colonization is being expanded, since that is more ambiguous. Instead, it can often rely on the way that diplomacy plays out, and the realpolitik of the situation.

The Monroe Doctrine is often taught as a policy that precluded European involvement in the western hemisphere. In fact, however, it precluded further colonization or interference with states in the western hemisphere. The Falklands were initially colonized before America was even founded so it was not further colonization or interference with an American state. The war in the Falkland Islands began because Argentina tried to take an existing British possession. The British were simply defending that existing possession, not acquiring a new one or interfering with an American state.

tl;dr: The war in the Falkland Islands doesn't fall within the purview of the Monroe Doctrine.

Palmerston's oft-cited remark about alliances can be generalized to apply to foreign policy "doctrines," too.

I say that it is a narrow policy to suppose that this country or that is to be marked out as the eternal ally or the perpetual enemy of England. We have no eternal allies, and we have no perpetual enemies. Our interests are eternal and perpetual, and those interests it is our duty to follow.

In other words you should look first to the durable interests that the doctrine was intended to uphold, and only second to the literal phrasing of the doctrine itself.

In this case, the underlying interest was to prevent a European great power from becoming a regional competitor to the United States. But Britain in 1982 was America's closest ally, lacked the means or ambition to become a regional power, was fighting purely defensively, and was fighting more to uphold British prestige and demonstrate British willingness to defend its territory than to secure some kind of strategic advantage (the Falklands having become more or less meaningless, strategically, once coal-burning warships were phased out.)

It also might be worth mentioning that the Monroe Doctrine, while announced as an American policy applying to all European powers, is probably best understood as an *Anglo-*American policy meant to preserve a status quo that both powers found useful. Whatever the formal terms of the doctrine, applying it to a disruption of the status quo at the expense of Britain would have been a new and unexpected departure. (Compare the Venezuelan Crisis, where under pressure of jingo sentiment, Cleveland embroiled the United States in a UK-Venezuelan boundary dispute, only to quietly push the matter off to an arbitration committee that eventually ratified the British position.)