The advert is a response to an open letter from Argentina’s President Cristina Fernandez de Kirchner, which was printed in two British papers.
Britain’s Prime Minister David Cameron has insisted the islanders must decide their own future.
Argentina invaded the islands in 1982 but was driven out by British forces.
A referendum on the islands’ political status is to be held in March.
On Thursday, President Cristina Fernandez de Kirchner published an open letter to David Cameron, in the Guardian newspaper and the Independent, repeating calls for the islands – which are known as the Malvinas in Argentina – to come under the sovereignty of her nation.
Cristina Fernandez de Kirchner urged David Cameron to abide by a 1965 UN resolution to “negotiate a solution” to the dispute.
But The Sun responded by taking out an advert in the Buenos Aires Herald – an English-language paper with a circulation of around 20,000 – telling Argentina to keep its “hands off”.
The advert refers to the 649 Argentines and 255 British servicemen whose lives were lost in the 1982 war and said it was a conflict fought to defend the principle of self-determination.
The ad goes on to dispute Argentina’s claim to the islands and points out British sovereignty dates back to 1765.
It ends with the words: “Until the people of the Falkland Islands choose to become Argentinean, they remain resolutely British.”
But the journalist Daniel Schweimler, who lives in Argentina, said The Sun’s message would not go down well.
Daniel Schweimler, who is based in Buenos Aires, said: “I’ve been here seven years now, and have never come across an Argentine who doesn’t believe that the Falklands belong to Argentina.”
“There’s never been any animosity towards me when I say I’m British, but I think it’s fair to say that almost across the board in a country of 40 million people that Las Malvinas, the Falklands, belong to them,” he added.
Argentine journalist Celina Andreassi agreed and says The Sun’s advert was quite provocative.
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