If there's a hill the UK is prepared to die on this is it, 99.9 percent of Falkland Islanders voted to remain part of the UK in a UN observed open election subsequent to the UK retaining control of the islands.
Argentina has never maintained any form of civilian population there in nearly 150 years, they want the oil and fishing rights.
First 99.9% then 99.8%? So, the support for a Falkexit is rising! Only give it a few hundred years and those Falklandiands are ready to join Argentina!
Interestingly, those islands are on argentinean sea. There're international laws that would say they are argentinean land... Even if they were occupied since maybe even before the two English invasions on Buenos Aires, a few centuries ago.
Bad news, though: I just found out that large swathes of southern England are in France's territorial waters, and the entire city of Manchester is owned by Liverpool for the same reason.
Territorial waters extend to a maximum of 12 nautical miles off the shore. States can claim a contiguous zone to up to 24 nautical miles, in which they can exercise some control to protect their territory and customs.
The exclusive economic zone (EEZ), in which a state controls all economic resources, extends to 200 nautical miles off the shore. These are not territorial waters.
All of the above have exceptions where the territorial waters and EEZs of two states would overlap.
The Falklands are 260 nautical miles off the Argentinean coast. So even if those 'international laws' you're so vaguely citing existed, they would not apply here.