The 46-year-old has faced widespread criticism for kissing player Jennifer Hermoso on the lips during the awards ceremony after Spain’s 1-0 victory over England in the Women’s World Cup final on 20 August.
In his statement, he said: “I made some obvious mistakes, which I sincerely regret from the bottom of my heart.
“The spontaneity and happiness of the historic moment led us to carry out a mutual and consented act, the product of great enthusiasm.
“At no time was there any aggression, indeed, there was not even the slightest discomfort, but an overflowing joy in both.
“I repeat: with the consent of both parties, both in the affectionate hugs, as well as in the peak and subsequent farewell full of affectionate mutual gestures, that occurred on the medals delivery stage.
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“I have also felt the growing support of people on the street and on social media.”
Earlier, the coach of the Spanish men’s national football team, Luis de la Fuente, apologised for applauding Mr Rubiales during the speech in which he refused to resign.
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Mr Rubiales had claimed in the speech last Friday that he was the victim of a “social assassination”.
The men’s coach was sitting in the front row during Mr Rubiales’ speech and clapped when he attacked “false feminism”.
Mr de la Fuente said the football federation president’s defiant speech was a “situation that overwhelmed many of us for which I was not prepared for” – saying he expected Mr Rubiales to resign.
He added: “I want to take this moment and my place as national men’s coach to explain the situation.
“I have received harsh criticism for it. Do you know what I think of the criticisms? They are entirely deserved, that I am sorry, that I understand them and ask forgiveness for them.”
Meanwhile, a Spanish government legal panel handling sports issues opened a formal case against Mr Rubiales.
Spain is hoping to remove him independently of a FIFA process that has already suspended him provisionally.
The government panel will decide if he abused his authority by kissing Hermoso or damaged the image and reputation of Spain at a sporting event, as the government claims.
He could be banned from office for two years.
“When the eyes of the world were on our players, his acts caused damage to our sport and our country that is difficult to repair,” said Miquel Iceta, Spain’s acting minister of culture and sport.