David Cameron clashed with Argentinian president Cristina Fernandez de Kirchner as the two countries marked 30 years since the end of the Falklands war.
Ms Kirchner used an address to a UN committee to demand talks over the sovereignty of the islands.
But the Prime Minister told a reception in London that there would be "no negotiation" - and warned that Britain would defend the territory by force again if necessary.
"Freedom is only won, and peace is only kept, because there are exceptionally brave people willing to travel to the other side of the world and lay their lives on the line," he said.
"This is not some game of global Monopoly, with nations passing a territory between them. It's about the islanders determining their own future."
Addressing the UN Committee on Decolonisation in New York last night, President Kirchner said distance undermined the UK's argument.
"How can it be claimed that, 14,000 kilometres away, that it can be part of the British territory?" she said.
"The UK is benefiting from its privileged position as a permanent member of the security council of the United Nations."
The comments came after war heroes and those they freed paid a series of moving tributes to the fallen.
Around 400 islanders gathered next to Liberation Memorial in Stanley yesterday to thank those who ended the 74-day occupation of the remote British Overseas Territory.