Isn’t Britain violating our Monroe Doctrine by holding the islands?
The simple reply to the conumdrum is that the Monroe Doctrine was never intended to be whatever the opposite of "retroactive" is (prospective?).
There is no independent clear assessment of the status of the Falklands, indeed, of the very existence of "Argentina" as an independent stable state in 1823.
The territory that ultimately became Argentina (and a few other current countries) was in a constant state of civil war or isurrections the entire first half of the 19th Century.
What finally became Argentina didn't even exist when the Monroe Doctrine was established.
Who cares?.
It was/is a policy of the United States. A policy, not a law.