The incident sparked outrage within and outside Spain, with many, including government ministers, players and coaches, demanding his resignation and forcing the federation to call an emergency assembly.
“I will not resign,” Rubiales repeated four times in quick succession at the meeting on Friday.
He also blamed “false feminists” for making him the victim of a “social assassination”.
“I apologise for my mistakes,” he added. “I'm going to continue fighting as my parents and my coaches taught me. Those who know me, know that we are going to get to the end.”
The 46-year-old is the subject of FIFA disciplinary proceedings after also grabbing his crotch in celebration of Spain's victory over England, while stood metres away from Spain's Queen Letizia and her teenage daughter Sofia in the stadium VIP area.
Hermoso has said in statement that such acts “should never go unpunished”. Rubiales made an apology deemed by many to be half-hearted on Monday.