The incident - which happened as Luis Rubiales handed the women's team gold medals after they beat England 1-0 in the final on Sunday - sparked outrage within and outside Spain, with many, including ministers, demanding Rubiales' resignation.
"We've seen his apology and that's not enough, he must be much clearer and convincing in apologising," Sanchez told a news conference. "He must take more steps to clarify a behaviour that is unacceptable."
Sanchez said the federation was not part of the Spanish government, which lacked the power to appoint or fire the federation's president.
Hermoso's first reaction to the kiss, planted on her lips as Rubiales cupped her face with his hands in post-game celebrations, was a comment to her teammates in the locker room.
"Hey, I didn't like it," she said, according to video footage posted on Instagram and YouTube by several media outlets.
However, it was later downplayed in a statement under her name issued through the federation.
As criticism mounted, Rubiales issued a video apology late on Monday, after initially calling critics "idiots".
"Surely I was wrong, I have to admit," Rubiales said in the video statement sent by the federation. "It was without bad faith at a time of maximum effusiveness."
Gender issues have become a hot topic in Spain over the past few years. The Socialist-led coalition government has presided over a raft of legal reforms, including around equal pay, abortion, sex work and transgender rights.
However, a sentencing loophole in a sexual consent law allowed some rapists to be released from jail early, resulting in a haemorrhage of votes for the hard-left party Podemos - in charge of the Equality Ministry that drafted the law - in local elections in May.