Ms Gibson disclosed the information after she resigned from her post
The Beckhams have received a High Court apology from their former nanny over her disclosure of private and confidential information about them.
Abbie Gibson has now agreed to give "permanent undertakings of confidentiality" to David and Victoria Beckham, Mr Justice Eady heard.
The court proceedings against Ms Gibson followed the publication of a newspaper article in April 2005.
She worked for the couple between May 2003 and March 2005.
After she resigned she gave an interview to the News of the World, in which she disclosed private and confidential information about the Beckhams and their family, the court heard.
The article was captioned "Beckhams behind closed doors".
The Beckhams' solicitor, Gerrard Tyrrell, said Ms Gibson "now accepts that she should not have breached her duties of confidence to David and Victoria Beckham and their children by speaking to the News of the World".
He said: "Ms Gibson has also agreed to the withdrawal of her claim for constructive dismissal that she made against the claimants which was issued in the Employment Tribunal in August 2005."
Ms Gibson "has unconditionally apologised to David and Victoria Beckham and their family for breaching her duties of confidence," he added.
Mr Justice Eady was also told that, on July 10, 2005, The People newspaper published an article entitled Exclusive: Beckham's Hate Calls To Nanny, which falsely stated that David Beckham had made a number of insulting and threatening telephone calls to Ms Gibson.
Mr Tyrrell added: "The People have already publicly apologised for making this false and defamatory claim and have paid damages to David Beckham.
"Ms Gibson is happy to confirm that David Beckham did not at any stage make any such telephone calls to her. She apologises if anything she said to The People gave them a false impression that such calls had been made."
Niri Shanmuganathan, solicitor for Ms Gibson, told the judge: "Ms Gibson wishes to use this opportunity to confirm that Mr Beckham has not made any rude or threatening telephone calls to her."
She was "happy to give the undertakings of confidentiality" detailed in an order before the court.