The Portugal forward will miss Real's league games against Villarreal, Getafe and Elche but he will be available for the King's Cup semi-final, first leg at home to holders Atletico Madrid later on Wednesday and the return game next week.
Ronaldo, who is celebrating his 29th birthday on Wednesday, was dismissed with 15 minutes left at the San Mames after pushing Bilbao defender Carlos Gurpegui in the face and confronting midfielder Ander Iturraspe.
He was banned for one match for the straight red card for violent conduct and had an extra two games added to his punishment for a gesture of dissent made while leaving the pitch, the RFEF said.
He was also fined 1,200 euros ($1,600) and Real, who have 10 days to appeal the ruling, were fined 1,050 euros.
Ronaldo is Real's top scorer with 22 goals in La Liga this season and in January won the FIFA Ballon d'Or award for the world's best player. ($1 = 0.7402 euros)
(Reporting by Iain Rogers, editing by Ed Osmond)
