What would have happened if peace wasn t signed after the UK took back the Falkland Islands?

Once the U.K. took back the Falkland Islands, the Argentine government had no remaining means to pursue the war - they couldn't recapture the islands and their air force had already taken severe losses. The Argentines could send what planes they had left to conduct aerial attacks on the Falklands and on British ships, but that would only end with the destruction of the Argentine Air Force; a similar story would be played out if Argentina used its navy for the same purpose.

Britain would certainly not attempt an invasion of Argentina - they didn't have the available manpower or naval power. Britain would basically sit on the Falklands and conduct naval patrols to seek out any Argentine military vessel that threatened the British positions, but otherwise would do nothing militarily - instead, Thatcher would lean heavily on Argentina's neighbors (along with the United States) to get them to pressure Argentina into coming to the negotiating table. After all, Britain's goal was purely to keep the Falklands - they had no interest in Argentina itself, and thus had no reason to escalate the war.

The UK could not have invaded Argentina. Many military experts considered even retaking the Falklands to be an almost impossible task. The fact that we did was a spectacular achievement for our troops. Forget invading and conquering the entire country. The British Armed Forces had some major limitations that made it impossible. Argentina was really in no place to take the Falklands back either. The idea of neither side wanting peace isn't realistic.

A stalemate such as on the Korean peninsula today. Neither peace nor war. The Falklands becoming a fortress, naval base and RAF station. The UK not wanting to attack Argentina but Argentina incapable to taking the Falklands.

Yes. The French claim we were threatening to go nuclear if they did not release the secret information about the arms they had sold Argentina. So they did. It was a very dangerous tme during the cold war, and we could not allow the soviet union to take advantage of the British forces being tied up in the South Atlantic indefinitely.

United nations. NATO and world politics would have taken over and sorted it out.

What-if questions are a waste of time.