Pressure has been growing on Luis Rubiales from many quarters
Rubiales, 46, has been widely criticised after the incident following Spain's Women's World Cup final win.
Hermoso, 33, said the kiss on her lips was not consensual.
Earlier, Spanish prosecutors opened a preliminary investigation into whether the incident amounts to a crime of sexual assault.
In other developments on Monday, Rubiales' mother Angeles Bejar has locked herself in a Spanish church and gone on a hunger strike in protest of the treatment of her son. In the evening, hundreds of people gathered in the centre of Madrid in protests called by feminist groups in support of Hermoso and against Rubiales.
Rubiales was suspended on Saturday by world football's governing body Fifa, a day after he had insisted he would not resign and insisted the kiss was "mutual and consensual".
"After recent events and the unacceptable behaviour that has seriously damaged the image of Spanish football, the presidents request that Luis Rubiales resign immediately as president of the RFEF," the federation (RFEF) said in a statement on Monday, after an extraordinary meeting of its regional presidents.
The presidents also gave their "unanimous support" to interim RFEF chief Pedro Rocha and called for an overhaul of the federation's management.
"We will urge the relevant authorities to carry out a profound and imminent organisational restructuring in strategic positions of the federation in order to make way for a new stage in the management of Spanish football," the statement added.
When Fifa announced Rubiales had been suspended on Saturday, it ordered him, the RFEF and its officials and employees not to attempt to contact Hermoso, whom the RFEF had threatened with legal action earlier that day.
Meanwhile, Spain's Sports Tribunal (TAD) also met on Monday to discuss the Spanish government's request to suspend Rubiales, with a decision yet to be announced.
Protestors applaud during a demonstration called by feminist associations in support of Spain's midfielder Jenni Hermoso
-- Courtesy of BBC Sport