Buckingham Palace has said it is disappointed that footage from 1933 showing the Queen performing a Nazi salute has been released.
The Sun has published the film which shows the Queen aged about seven, with her mother, sister and uncle.
The palace said it was "disappointing that film, shot eight decades ago... has been obtained and exploited".
The newspaper has refused to say how it got the footage but said it was an "important and interesting story".
'Misleading and dishonest'
The black and white footage, which lasts about 17 seconds, shows the Queen playing with a dog on the lawn in the gardens of Balmoral, the Sun says.
The Queen Mother then raises her arm in the style of a Nazi salute and, after glancing towards her mother, the Queen mimics the gesture. Prince Edward, the future Edward VIII, is also seen raising his arm.
The footage is thought to have been shot in 1933 or 1934, when Hitler was rising to prominence as Fuhrer in Germany but the circumstances in which it was shot are unclear.
A Palace source said: "Most people will see these pictures in their proper context and time. This is a family playing and momentarily referencing a gesture many would have seen from contemporary news reels.
"No-one at that time had any sense how it would evolve. To imply anything else is misleading and dishonest."
The source added: "The Queen and her family's service and dedication to the welfare of this nation during the war, and the 63 years the Queen has spent building relations between nations and peoples speaks for itself."
BBC Royal correspondent Sarah Campbell said Buckingham Palace was not denying the footage was authentic but that there were "questions over how this video has been released".
Who was the man in the video?
- Edward was uncle of the young princess Elizabeth and brother of George VI
- He briefly became King himself in 1936 but abdicated just 326 days later because of his plans to marry American divorcee Wallis Simpson - a marriage government and church figures deemed unacceptable
- Replaced by George VI, Edward was one of the shortest reigning monarchs in British history
- In October 1937, Edward and his wife - by now the Duke and Duchess of Windsor - visited Nazi Germany with the idea of discussing becoming a figurehead for an international movement for peace on Hitler's terms
- During the controversial visit they met Hitler and dined with his deputy, Rudolf Hess
- Evidence emerged Edward went to the early stages of a concentration camp, although it is not thought evidence of mass murder was made clear to him
- He moved to France with the Duchess after the war and died there in 1972
Dickie Arbiter, a former Buckingham Palace press secretary, said the Palace would be investigating.
"They'll be wondering whether it was in fact something that was held in the Royal Archives at Windsor, or whether it was being held by the Duke of Windsor's estate," he said.
"And if it was the Duke of Windsor's estate, then somebody has clearly taken it from the estate and here it is, 82 years later.
"But a lot of questions have got to be asked and a lot of questions got to be answered."
Sun managing editor Stig Abell said he did not accept Buckingham Palace's accusation that the footage had been "exploited".
He said the newspaper had decided to publish the story because it was of great public importance and the involvement of Prince Edward gave it "historical significance".
The then Prince of Wales faced numerous accusations of being a Nazi sympathiser and was photographed meeting Hitler in Munich in October 1937.
BBC royal correspondent Peter Hunt
It's an arresting, once private image on the front of a national newspaper.
Its publication has prompted Palace officials to talk about a breach of privacy and the Sun to argue it's acting in the national interest.
Apart from the obvious anger on one side, it's striking how both sides have talked of the need to put the home movie in its "proper context".
From the Palace perspective this is a six-year-old princess who didn't attach any meaning to the gesture. Such an explanation doesn't, of course, explain the thinking of her mother.
Those around the royals are also keen to focus on the war record of the then King, Queen and their two daughters.
What they're less keen to focus on - and what the Queen would like not to be reminded of - is the behaviour of her uncle.
A man, who was briefly King, and whose fascination with Nazi Germany is well documented.
Mr Abell said: "We are not using it to suggest any impropriety on behalf of them. But it is an important and interesting issue, the extent to which the British aristocracy - notably Edward VIII, in this case - in the 1930s, were sympathetic towards fascism.
"That must be a matter of national and public interest to discuss. And I think this video and this footage animates that very clearly."
Mr Abell told the BBC the video was a piece of "social history" and said the paper had set out the context of the time and explained that the Queen and Queen Mother went on to become "heroes" of World War Two.
He denied the video had intruded into the Royal Family's privacy.
"I think this is a piece of social history. One of the most significant events in our country's history, the Second World War, the rise of Nazism, one of the most pernicious movements in human history, and I think one is entitled to have a look at some of the background to it."
He added: "We're very clear. We're of course not suggesting anything improper on behalf of the Queen or the Queen Mum."
The Queen was 13 when World War Two broke out and she later served in the Women's Auxiliary Territorial Service.
In June she made a state visit to Germany where she visited the Bergen-Belsen concentration camp and met some of the survivors and liberators.