If a Loyalist area of one of the 13 Colonies had held a referendum in 1776 and resolved to stay British, then the American Republic would not have recognised that, either.
There is no more reason to expect the United States to support an old European imperial power over the Falkland Islands than there is to expect any old European imperial power, no matter how Latin or Catholic or whatever, to support Argentina. France and Portugal gave us invaluable support in 1982. Well, of course. America, er, didn't... Well, of course not.
Latin America now matters vastly more to American foreign policy than anywhere in Europe does, and now matters in American domestic elections in a way which Britain, in particular, simply never has done. Except as a bogeyman from time to time, by no means only in the distant past by American standards.
Driving out British influence from the Americas was the whole point all the way back at the time of the Declaration of Independence. Extended from the hemisphere to the globe, there has never been a break in American pursuit of that one overriding objective, to which any and everything else has been subordinated as and when necessary. Nor will there ever be.
Will all British Citizens living on the Falkland Islands, including the Saint Helenians working there, get to vote in the referendum? If so, then even if it had not been in the bag before, then it certainly is now. God Save The Queen.