Hector Timerman claimed there are only British citizens living on the disputed islands adding the United Nations only acknowledges two parts in the conflict - the UK and Argentina.
It came a day after he told MPs that Buenos Aires would not recognise the result of a referendum of the islanders next month on whether they wish to remain part of the British overseas territories.
He also said in a newspaper interview that the islands would be under Argentine control within 20 years, while denouncing the British as "fanatics".
The Falklands Islands parliament has written to Timerman warning him "no amount of harassment and intimidation" would change their minds that they do not want to be ruled from Buenos Aires.
Speaking at a press conference in central London today, Timerman said: "The Falklands islanders do not exist. What exists is British citizens who live in the Islas Malvinas.
"The United Nations does not recognise a third party in the conflict. It says there are just two parts - the UK and Argentina."
Legislative assembly members Jan Cheek and Dick Sawle were at the Foreign Office for the discussion and said afterwards they were "disappointed but hardly surprised" at his absence.
"It is, and must always be, for them to decide their own future," Hague said afterwards, affirming the Government's full support for the referendum and the islanders' right to choose.
Cheek said: "Timerman dismisses us as 'settlers'. Well, we are settlers. Like countries across the continent of the Americas, we came into existence through waves of settlement from Europe and elsewhere.
"Indeed, we Falkland islanders settled in our home long before many parts of Argentina were settled by the Argentines.
"Timerman knows full well that it is simply untrue to say that we have fewer political rights than anyone else, or that the United Nations has said that self-determination does not apply to the Falklands, or that UN resolutions preclude Argentina from sitting down with us.
"Repeating these misrepresentations doesn't make them any truer, however inconvenient for Argentina."
Buenos Aires was "deeply worried about our referendum", she said, "which is why they spend so much time dismissing it".
"Talk of the Falklands being Argentine in 20 years makes for good headlines - but smacks of desperation.
"We are a thriving community, with a growing international voice. We want good neighbourly relations with Argentina, we are willing to talk with them and we cannot be ignored. Self-determination is a universal right.
"We are very grateful to the British Government for standing up for our freedom, in contrast to those who would seek to deprive us of it.
Hague said: "I welcome their forthcoming referendum in March. This will be a chance for the people of the Falkland Islands to express their views about how they wish to be governed, and in a democratic and incontestable way without other people speaking for them. They have the British Government's full support."
He said there was "no way" he could have excluded the islanders' representatives from the discussions, "especially given the current Argentine government's behaviour towards the islanders".
Timerman told reporters at the Argentine ambassador's residence in London that he had travelled to the UK for a gathering of representatives from European Union countries that are in favour of dialogue.
Referring to his refusal to meet Hague, he said: "In an act of courtesy I offered a meeting to Foreign Secretary William Hague so we could discuss bilateral topics and multilateral issues like the UN Security Council.
"The attempt to impose a condition and an ultimatum made me reject the possibility of a meeting.
"With that attitude, he prevented us from meeting to discuss all the issues relevant to the bilateral relationship."
Timerman invited the Foreign Secretary to go to Buenos Aires for a meeting "without ultimatums or conditions, as two friendly countries which are simply seeking dialogue".
Asked if the conflict over the islands was "sterile", Timerman said: "We are talking about the territorial integrity of a country that was invaded four times by the UK and defeated three times - but one time they were able to divide our territory.
"It is not a sterile matter, it is a very important issue for us. Maybe it is sterile for the UK, but not for us."
Timerman also said Argentina was not seeking the UK's "surrender" and had made attempts to improve the lives of islanders with measures such as setting up direct flights between the islands and Buenos Aires.
"Argentina is looking for a dialogue to pacifically resolve a dispute over sovereignty," he said.
"As you know, there are very few inhabitants of the Islas Malvinas who were born in the Islas Malvinas.
"But for us, they are Argentinian citizens, and as Argentinian citizens they have all the social, civil, economic, political rights as any Argentinian citizens born in continental Argentina."
He added: "The UN is very clear when it says that self-determination is only for native peoples and not for implanted populations."
Timerman said the Argentine government would continue to seek to impose sanctions on British oil and gas companies which explore the waters of the South Atlantic.
"They are stealing part of the natural resources that belong to the Argentinian people," he said.
Britain believes the discovery of oil and gas around the islands is one of the reasons Argentina has been ratcheting up its rhetoric over the territory.