Rubiales has been under fire after kissing Hermoso on the lips while handing the women's team their medals after they beat England 1-0 in the final on Sunday.
The incident sparked outrage within and outside Spain, with many, including government ministers, demanding his resignation.
Pressure continued to mount during the week after FIFA opened disciplinary proceedings against Rubiales and Hermoso said in a statement that such acts "should never go unpunished."
As backlash mounted, Rubiales, who initially called his critics "idiots", issued a video apology late on Monday, but it failed to quell the uproar.
An RFEF spokesperson declined to comment
Rubiales, who played for several lower league clubs in Spain, finished his playing career in 2009 with Hamilton Academical in Scotland.
A year later he was named President of the Association of Spanish Footballers (AFE), a position he held until announcing his candidacy for the RFEF top job in 2017. He was elected in May 2018.
He promised to modernise the structure, increase turnover and make the federation more transparent, following the arrest of former president Angel Maria Villar on corruption charges.
However, Rubiales' mandate was full of controversies, including the shocking decision to sack Spain manager Julen Lopetegui two days before their opening 2018 World Cup match and accusations of wrongdoing over a multimillion-euro deal to relocate the Spanish Super Cup to Saudi Arabia.
There was also a mutiny within the Spain women's squad last September when a large group of players demanded that long-standing coach Jorge Vilda be sacked.
The RFEF backed Vilda and he cut 12 of the 15 players involved in the dispute from his World Cup squad.