In the spirit of Manifest Destiny.
So let's say that rather than just destroy the settlement, the US decides that it would like to keep the Falklands as a convenient seal hunting/trading port in South America of its own. Argentina really hasn't the ability to take it back at this time, and whenever the British sail in they find that it's still occupied by the Americans, who never get around to leaving.
What goes on afterwords?
Reading the wikipedia entry on the pathetic history of the original Falklands conflicts is enlightening. France, Britain, Spain, and then Argentina kept laying claim to the islands, and then leaving them because there was basically not much there. After a peace treaty in 1771, Spain and Britain both had colonies on the islands. But Britain abandoned them in 1776 and Spain in 1811 (neither relinquishing claim). When the Argentines declared independence in 1816, they declared that the islands were now theirs since they'd been administered from Buenos Aires while a Spanish colony, but they did nothing but plant a flag until 1826 at which time they established a settlement and penal colony.
The settlement was actually destroyed by the American navy in 1831, after the islands' governor seized US ships over a dispute about seal hunting rights.
Then in 1833 the British basically waltzed in and took over. And that's the history of the grand Argentine historical claim to the Falkland Islands.
So let's say that rather than just destroy the settlement, the US decides that it would like to keep the Falklands as a convenient seal hunting/trading port in South America of its own. Argentina really hasn't the ability to take it back at this time, and whenever the British sail in they find that it's still occupied by the Americans, who never get around to leaving.
What goes on afterwords?