The newspaper said the Queen vented her anger with Brussels at the pro-EU former deputy prime minister Nick Clegg during a lunch at Windsor Castle in 2011.
In a statement, The Sun said: “The Sun stands by its story, which was based upon two impeccable sources and presented in a robust, accessible fashion. The Sun will defend this complaint vigorously.”
A Buckingham Palace spokesman 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 1 of the Editors’ Code of Practice.”
Clause 1 in the code relates to accuracy and states: “The press must take care not to publish inaccurate, misleading or distorted information or images, including headlines not supported by the text.”
It requires that “significant inaccuracy, misleading statement or distortion must be corrected, promptly and with due prominence, and - where appropriate - an apology published”.
The Sun’s front page headline read: “Queen backs Brexit” and the 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”.
Former Liberal Democrat leader Mr 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.”
The rare move by the Palace illustrates the frustration within the Royal Household at the Queen being drawn into a political row.
This is the first time a complaint has been registered by the Palace about or on behalf of the Queen with the Independent Press Standards Organisation, the regulator of the newspaper and magazine industry, which was set up in 2014.