Buckingham Palace complained about the story published in March, which detailed an occasion when the Queen allegedly vented her anger with Brussels at the strongly pro-EU Nick Clegg when he was deputy prime minister.
The verdict from the Independent Press Standards Organisation (IPSO) found the newspaper's headline had breached Clause 1 (accuracy) of the Editors' Code of Practice.
The Sun's editor-in-chief Tony Gallagher said on Wednesday that he respected IPSO's ruling, but did not believe that the headline was inaccurate.
He told BBC Radio 4's Today programme: "I don't accept that we made an error at all. We made a judgement that the headline was right and that it was backed up by the story.
"We knew more than we put in the public domain. The sources were so impeccable that we had no choice but to run the story in the way that we did."
Gallagher, who approved the headline before it went to print, said that he would not do anything differently if the same decision had to be made today.
He said that it was "only when it became apparent that the story was going to become exonerated that the palace narrowed its claim to the headline".
The journalist added that the headline was "only misleading if you exclude the words 'bombshell claim over Europe vote'".
The Rupert Murdoch-owned paper quoted a "senior source" as saying that people who heard their conversation "were left in no doubt at all about the Queen's views on European integration".
The article said two unnamed sources had claimed that the Queen made critical comments about the EU at two private functions - first with Clegg at a lunch for Privy Counsellors at Windsor Castle in 2011, and at a reception for MPs at Buckingham Palace.
At the time, Clegg dismissed the report as "nonsense", while the Palace said: "The Queen remains politically neutral, as she has for 63 years.
"We will not comment on spurious, anonymously sourced claims. The referendum is a matter for the British people to decide."
Gallagher told the Today programme that Clegg had a "number of opportunities" to deny the story before it went to print yet had not done so.
The former Liberal Democrat leader only denied the story "when it created a furor" and after he consulted with the palace, Gallagher said.
It is not the first time that The Sun has been found to have flouted press standards.
In March the tabloid was forced to admit that a story printed on its front page claiming that one in five British Muslims sympathised with so-called Islamic State (Isis) was “significantly misleading”.
IPSO received an unprecedented number of complaints following the front page article in November.
The press watchdog ruled that the publication misrepresented the results of a poll.
In the same month as IPSO's verdict, David Dinsmore, former editor of The Sun, was convicted of breaching the Sexual Offences Act after the paper printed a photograph of Adam Johnson’s teenage victim, who should have been guaranteed anonymity.
The Sun published a photograph of the girl following the former Sunderland and England player’s arrest in March last year.
In Wednesday's edition, The Sun published IPSO's ruling on page 2.
The Sun article read: "IPSO acknowledged the importance of headlines in tabloid newspapers.
"However, it did not follow from the comments the article reported that the Queen wanted the UK to leave the EU as a result of the referendum: that suggestion was conjecture and the committee noted that none of those quoted in the story were reported as making such a claim.
"The headline was not supported by the text.
"It was significantly misleading - given that it suggested a fundamental breach of the Queen's constitutional obligations - and represented a failure to take care not to publish inaccurate, misleading or distorted information in breach of clause 1.
"The complain under cause 1 was upheld."
IPSO said that, while the article itself did not breach the code, the headline did as it was "a factual assertion that the Queen had expressed a position in the referendum debate, and there was nothing in the headline, or the manner in which it was presented on the newspaper's front page, to suggest that this was conjecture, hyperbole, or was not to be read literally".