Rubiales, 45, kissed Hermoso on the lips after she collected her medal, provoking an immediate global reaction.
"Certainly I made a mistake and I have to acknowledge that," Rubiales said in a video posted on social media by Spanish television.
"It was done without any ill intention in a moment of the highest exuberance. Here we saw it as natural and normal but outside it has caused a commotion."
"I have no choice but to apologise and to learn from this... and when representing the federation take more care," he said, adding that he thought the furore was "idiotic".
Earlier Spain's sports minister on Monday had demanded that Rubiales apologise.
"I think it is unacceptable to kiss a player on the lips to congratulate her," acting minister Miquel Iceta told Spanish public radio.
"The first thing he has to do is to give explanations and make apologies, it is the logical and reasonable thing to do," Iceta said.
The minister added that while the 1-0 victory over England on Sunday in Sydney was "a moment of intense emotions", public officials "have to be extremely careful because we are giving a message to society and the message is equal rights, it is respect".
'Spontaneous'
Rubiales on Monday stressed that he had a "magnificent relationship" with Hermoso.
On Sunday, the player posted a video on Instagram showing the celebrations in the changing room in which she responded to teasing from team-mates by saying: "I didn't like it, eh!" while laughing.
Later on Sunday the RFEF released their own quotes from Hermoso.
"It was a totally spontaneous mutual gesture because of the immense joy that winning a World Cup brings," Hermoso was quoted as saying.
"The president and I have a great relationship. His behaviour with all of us has been outstanding and it was a natural gesture of affection and gratitude.
"A gesture of friendship and gratitude cannot be gone over so much, we have won a World Cup and we are not going to deviate from what is important."
Spain's preparations for the World Cup were thrown into disarray last September when 15 players sent an email to the RFEF saying they did not want to be considered for selection.
Their protest was aimed chiefly at coach Jorge Vilda and his methods, including complaints that he was too strict.
Rubiales supported Vilda and only three of the 15 were recalled for the World Cup.
