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.