The Argies dont do things legally

Britain signed what is called the “Optional Clause” decades ago – long before the Falklands War in 1982. This clause allows any country to bring a case against Britain at the International Court of Justice (ICJ) about any subject. This would be well-known to international lawyers in Argentina. So they must have been reluctant to go to the ICJ for fear of losing.

Argentina has been criticised for going to war without first going to the ICJ. Argentina did agree to accept the result of the arbitration with Chile over the disputed Beagle Channel islands, but then reneged on this when it lost and rejected the result.

Also Britain invited Argentina to go to the ICJ several times over the Falklands Dependencies (particularly South Georgia) because of Argentine harassment there. Britain eventually tried to take the case to the ICJ unilaterally, but Argentina refused.

MORE PROOF THAT THE FALKLANDS ARE BRITISH

This map is known as the 1882 Latzina Map, and it was financed by the Argentine Foreign Ministry and published 120,000 copies that were distributed to Argentine consulates all over the world to attract immigrants to Argentina.
In 1977 the International Court of Justice ruled that this map was evidence in a dispute over the Beagle Channel islands (3 small islands at the bottom of the map) between Chile and Argentina.
The International Court of Justice ruled that the islands belonged to Chile mainly thanks to the fact that this very map marked the Argentine mainland in a brownish colour and the beige colours marked places outside Argentina's mainland like Uruguay, Chile and even the Falklands.
This map was made after the "Convention of Settlement" - A Peace Treaty that was suppose to have ended all possible Argentine claims to the Falklands in 1850 and this map is evidence that Argentina did drop it's claims to the islands.
Argentina had gone silent over the Falklands after the Convention of Settlement, and in fact no Argentine maps printed between 1850 and 1884 clearly showed the Falklands as Argentine territory at all. Argentina simply went back on the treaty in 1884.

Not that the Argentine s will respect any of this , they are a rogue state ruled by a deluded female maniac.