While I do understand the Cold War strategic implications of not invoking the Doctrine, I see that it was selectively enforced (Cuba being a good example of when we used it when we liked what it did, according to Kennedy who said that Russian presence violated the Doctrine, more or less). My other question along with that would be if there was any debate about the Falklands with the Monroe Doctrine, because it seems exactly like the type of action its writers wanted to prevent.
The US was trying to convince Argentina and Britain to compromise by being pragmatic and avoiding discussions of sovereignty. Invoking Monroe Doctrine would not really have helped it to do that and could have inflamed opinion in both countries. The Falklands War was a disaster for US policy because it made them choose between two friendly powers. In the end they had to choose the UK because of its strategic importance as the US's key European ally.
Leaving the Monroe Doctrine aside, the US could in theory have tried to exert its power and prevent the war but there would have been massive damage to its relationship with the UK at a crucial point in the Cold War - including the risk of Labour taking power at a time when it was committed to nuclear disarmament.
[Edit : source added] this report on the release of cabinet papers shows how close the UK were to the US. http://www.theguardian.com/uk/2012/dec/28/thatcher-falkland-islands-national-archives
No, we could not have invoked the Monroe Doctrine. The doctrine was written to forbid further involvement in the Americas by European powers, but never forbade those powers already involved to continue their involvement. So, Britain could not establish new colonies in the Americas, but could administer and defend those it already had. As such, Britain had every right to maintain involvement in the Falklands, up to and including taking military action in response to invasion. The Monroe Doctrine only applied against European powers, so what Argentina did could not violate it. Furthermore, America had no interest in getting involved in the Falklands. Britain did not paricularly need our assistance, and we certainly weren't going to back Argentina over our biggest ally. That is why we stayed out of it for the most part. We just didn't have a dog in that fight.