The palace complained to the Independent Press Standards Organisation about Wednesday's report that claimed the Queen had voiced strong Eurosceptic views during a lunch with former deputy prime minister Nick Clegg.
Clegg attended a lunch at Windsor Castle in 2011, where, according to the tabloid, the monarch gave him a dressing down on the direction of the EU and European integration.
The Queen is meant to remain strictly neutral on political matters and the palace has complained, saying the story is inaccurate.
A palace spokesperson said: "We can confirm that we have this morning written to the chairman of the Independent Press Standards Organisation to register a complaint about the front page story in today's Sun newspaper.
"The complaint relates to clause one of the Editors' Code of Practice."
Clause one of the code relates to accuracy and says the press must not publish "inaccurate, misleading or distorted information".
In response, the paper said it "stands by its story", saying it was "based upon two impeccable sources and presented in a robust, accessible fashion. A spokesman added: "The Sun will defend this complaint vigorously".
The paper cited a "highly reliable source" who relayed details of the conversation but Clegg later said: "I've no recollection of this happening & its not the sort of thing I would forget."
The Sun on Wednesday
Tom Newton Dunn, The Sun's political editor, called this a "non denial denial," a term used by journalists to describe an attack on a story that does not actually refute its assertions.
Speaking on LBC Radio, he said: "He's not confirming it. He's not denying it in any way, shape or form. As slightly ruder people than me have already said today, it's one of those 'non-denial denials' we hear from politicians.
"A form of words quite carefully constructed to make you believe they're denying the story when they're actually not... We do believe this incident has happened. We do believe Her Majesty has strong views on Europe."
He said the palace and Clegg conferred before responding to his request for comment, prompting host Nick Ferrari, an ex-Sun journalist, to say: "It's got to be true... They don't ring each other unless they're nervous about something."
Nick Robinson, former BBC Political Editor, noted the article was based on a four-year-old comment about the direction of the EU, rather than a recent endorsement of a Brexit.
Patrick O'Flynn, who went to work for Ukip after quitting his job a Daily Express journalist, said the story was a "great for Brexiteers".
The Sun's source told the paper: "People who heard their conversation were left in no doubt at all about the Queen’s views on European integration.
"It was really something, and it went on for quite a while. The EU is clearly something Her Majesty feels passionately about."
In a separate piece, Newton Dunn wrote the paper would not have published the story if the comments "had not come from two different and impeccably placed sources".
Tom Newton Dunn on LBC
He also predicted the story would trigger "an almighty row". He told LBC: "The Queen is supposed to be above politics. Constitutionally, it's her government, whatever the government chooses to do, she has to sign and implement."
Clegg's spokesperson later added: "This is categorically untrue. Nick has no recollection of this conversation and it is not the sort of conversation you forget."
Labour has called for an inquiry into the source of the leak and written to Cabinet Secretary Sir Jeremy Heywood to ask how private conversations became public.
Justice Secretary Michael Gove, one of the most prominent figures backing Brexit in June's referendum, is being touted as the most likely figure to have leaked details.
A Labour spokesman said: “(MP) Wes Streeting is writing a letter. It will be interesting to see the response. Let’s see what comes back.”
After Prime Minister's Questions, the PM’s spokesman was asked repeatedly whether Cameron discussed the row with Gove this morning.
He said: “I wouldn’t guide you towards that conclusion.”