It's all down to posturing Rodo, had the two Governments started a quiet, civilized dialogue (something that was outwardly not on the Agenda, but there are ways) then the whole thing could be discussed, and, eventually, an agreement might have been reached. Instead, it boiled down to hurling insults from 200 years ago at the UK and calling in US actors to be their mouthpiece.
Personally, what I would have done is talked quietly to the US and said "Look, we want a dialog about the Malvina's, but the UK won't even consider it, if you can help us diplomatically then you might just get your foot in the door if any oil is found there". Sneaky, I know, but that's diplomacy for you. It lets the US play it's favourite role as 'World problem solver', it means the UK actually have to listen, and it means Argentina finally get their grievances heard for better or for worse.
@samiam, the thing is, the UK and Argentina have now both made this a matter of Pride, the political cost of backing down to the UK, particularly after some of the comments made about the UK in general over the past few weeks, would probably be considered worse than the financial cost of 'what it takes' to re-capture them if it happened.
That said, I don't think it would happen, Argentina was under a completely different, more militaristic Government back then, which was why the US backed the UK's right to the Falklands at the time.