David Cameron clashed with Argentina 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.
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.
A thick snow storm began as Falklanders, veterans and VIPs left a service of thanksgiving held at Stanley's Christ Church Cathedral.
Young and old packed into the church with standing room only at the back to take part in the commemoration to those who died in 1982.
The brief but bitter war ended on June 14 1982 as Argentinian commander General Mario Menendez surrendered to the British at Stanley.
The fighting cost the lives of 255 British servicemen, three Falkland Islanders and 655 Argentinian soldiers.
Islanders lined the road next to Stanley Harbour in which HMS Clyde has taken anchor for the commemorations.
Union flags were hung alongside the Falkland flag opposite Liberation Monument where an Act of remembrance was held.
About 30 veterans braved the cold to proudly lead a parade of servicemen from the Royal Navy, the Parachute Regiment, the Royal Air Force and the Falkland Islands Defence Force.
They were accompanied by the Salamanca band of the Rifles as they marched from the Cathedral to Liberation Monument.
Following the Last Post, a number of wreaths were laid at the foot of the memorial by veterans, politicians and islanders.
The service was attended by Foreign Office minister Jeremy Browne, who said: "I think it is hard to convey to anybody who isn't here, who is back home in Britain, just how much this means to the Falkland Islanders."
He said it was "humbling" and he was "very proud" to be there.
In a speech at the annual Falkland Islands Government reception last night, Mr Cameron paid tribute to the bravery of the then prime minister Baroness Thatcher and the Armed Forces.
"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.
"So to everyone who served in the Falklands - those who are here today and those who aren't - I say on behalf of the British people that we are proud of you and we salute you. We will always be in your debt."
The Prime Minister said the Falklands was enjoying economic growth and industries such as tourism and fishing were "thriving".
"There is only one shadow on the horizon. And that is the aggression from over the water," Mr Cameron said.
"We've seen the president trying to restrict the movement of Falklands vessels, banning charter flights to and from Argentina and today, escalating the debate at the UN.
"In the face of this I want to be absolutely clear - this Government's long-term goal for Latin America is not bickering and hostility, it is partnership.
"We've been expanding our missions, sending more ministers on trade visits, increasing the number of Spanish- and Portuguese-speaking diplomats.
"With Argentina in particular there are so many things we should be working together on - managing fish stocks, increasing trade, environmental issues. This is the kind of co-operation we need - and that's why we want to have a reasonable, sensible relationship with Argentina.
"But let me be equally clear on this - when it comes to the sovereignty of the Falkland Islands, there will be absolutely no negotiation.
"This is not some game of global Monopoly, with nations passing a territory between them. It's about the islanders determining their own future.
"This has been their home for almost 180 years. There are children whose ancestors have lived there for generations. The roots go deep, and they will not be ripped out."
Highlighting a planned referendum of Islanders on whether they wish to remain linked to the UK, he went on: "My message to the government of Argentina is this: the UK has no aggressive intentions towards you. Accusations of militarisation and nuclear threats are hyperbole and propaganda.
"But do not under-estimate our resolve. Threats will not work, attempts to intimidate the islanders will not succeed, because Britain stands ready and willing to stand up for the Falkland Islanders at any time.
"As long as they wish to remain a British territory, that is the way it will stay."
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 head of state said the dispute was a challenge to the international community to overcome outdated "prejudice and cliches".
She insisted Argentina just wanted to "talk" about the islands' sovereignty.
"Can someone in the modern world deny that possibility?" she added.Reuse content