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.
Sanchez said the federation was not part of the Spanish government, which lacked the power to appoint or fire the federation's president.
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.
Persons:
Pedro Sanchez, Royal Spanish Football Federation Luis Rubiales, Juan Medina, Jenni Hermoso, Luis Rubiales, Rubiales, We've, that's, Sanchez, Hermoso's, Inti Landauro, David Latona, Bernadette Baum
Organizations:
Soccer Football, FIFA, Spain's, Moncloa, Royal Spanish Football Federation, Rights, England, YouTube, Equality Ministry, Thomson
Locations:
New Zealand, Madrid, Spain, Spanish