And screw the people who actually live there, right?
A better way to think about this is to consider the case of Saint Pierre et Miquelon, which are two islands off the coast of Canada -- literally 19 km away, and so much closer than the Falklands are to Argentina. They are French territory, inhabited by French citizens. And (aside from occasional disagreements about who gets to explore for undersea resources in the vicinity of the islands) Canada really doesn't seem to have a problem with this, because (in this area, at least) Canada is a more mature nation than Argentina.
There's no such right and even if there was, it would lead to perverse situations such as Britain getting to claim Calais which is closer to London than Paris (94 miles versus 147!).
There are the Channel Islands, but perhaps you could claim that the Channel Islands were the ones to occupy England and not vice versa since they were part of Normandy!
Sorry, I was trying to satirise the argument for why it should be Argentinian, where the fundamental (but generally unsaid) reason is "because it's close to us".
And as we all know, national borders are decided by plotting capitals on a map and running a mathematical process to minimise distance from capital, right?