The Sun has placed an advert in a Buenos Aires newspaper telling Argentina to keep its "hands off" the Falkland Islands.
The move comes a day after Argentine president Cristina Fernandez de Kirchner called on Cameron to relinquish control of the Falklands in an advert published in the Guardian and Independent.
The Sun's advert says the 649 Argentinian and 255 British soldiers who died in the 1982 war fought to defend the principle of self-determination.
The ad, placed in English-language newspaper the Buenos Aires Herald, said the Falklands would remain "resolutely British" until the islanders chose otherwise.
The paper has a circulation of 50,000.
Argentine journalist Celina Andreassi said the Sun's advert is likely to be met with anger. Speaking to BBC 5Live she said: "People will wonder what has The Sun got to do with it."
The Sun has also come under criticism on social media, with accusations the paper has 'no business' conducting diplomatic relations.
A referendum on the islands status is due to be held in March.
Kirchner reignited the row after insisting in the advert that Argentina was forcibly stripped of the Malvinas - the Argentinian name for the islands - in "a blatant exercise of 19th-century colonialism".
However Cameron rebuffed claims by the Argentine president, saying the Falkland Islands is in the hands of the people who live there, not Argentina.
During a visit to Preston, the prime minister said Kirchner should "listen" to the result of a referendum to be held on the Island, and if the people chose to remain British they would have his "100%" backing".
"The future of the Falkland Islands should be determined by the Falkland Islanders themselves, the people who live there," he said.
"Whenever they have been asked their opinion, they say they want to maintain their current status with the United Kingdom.
"They're holding a referendum this year and I hope the president of Argentina will listen to that referendum and recognise it is for the Falkland Islanders to choose their future, and as long as they choose to stay with the United Kingdom they have my 100% backing."
Cameron and de Kirchner clashed over the Falklands when the pair came face to face at the G20 summit in Mexico last June.
He rejected her demand for negotiations over the sovereignty of the islands and told her to respect the result of a referendum, when the Falklanders will vote on whether they wish to retain their ties with Britain.
In December Argentina protested at Britain's decision to name a vast swathe of Antarctica Queen Elizabeth Land, with its foreign ministry handing a formal protest note to British ambassador John Freeman in Buenos Aires.
The area, which makes up around a third of the British Antarctic Territory, is also claimed by the South American country.
Barry Elsby, a member of the Islands' Legislative Assembly, said: "We understand that the Argentine government has put out a letter that both calls our home a colony and claims that the United Kingdom is ignoring United Nations General Assembly resolutions.
"We are not a colony; our relationship with the United Kingdom is by choice.
"Unlike the government of Argentina, the United Kingdom respects the right of our people to determine our own affairs, a right that is enshrined in the UN Charter and which is ignored by Argentina."