The newly-crowned FIFA Ballon d'Or winner was given a one-match suspension for the straight red card and handed a further two-match ban for a gesture he made as he left the field.
Ronaldo is now set to miss the league matches against Villarreal, Getafe and Elche, although Madrid, who play arch rivals Atletico Madrid in the first leg of the Copa del Rey semi-finals on Wednesday night, have 10 days to appeal against the decision.
Ronaldo, who turned 29 on Wednesday, was dismissed in the 75th minute of the 1-1 draw at San Mames following an off-the-ball incident first involving Athletic captain Carlos Gurpegi and then home midfielder Ander Iturraspe.
The former Manchester United man was pushed by Gurpegi after appealing for a handball in the area and responded by brushing his hand into the Athletic man's face.
Gurpegi tumbled to the turf and a melee between the teams ensued with Ronaldo and Iturraspe confronting each other.
Iturraspe ended up with a yellow card while Ronaldo was shown a straight red card - something his team-mates believe was undeserved, with Xabi Alonso saying after the match: 'It wasn't aggression, the red to Cristiano Ronaldo is excessive."
Ronaldo then courted more controversy while leaving the pitch as he was seen to slap his cheek several times, which is a mocking gesture in Spain to suggest someone has some nerve and was perceived as criticism of the officials' decision.
The Spanish football federation's competitions committee on Wednesday announced that Ronaldo has been handed a combined three-match ban - one match for the incident involving Iturraspe and two more for the gesture - plus a 1200 euro fine.
Ronaldo is the leading scorer in the Primera Division this season with 22 goals in 21 matches, while Madrid are third in the standings, three points behind leaders Atletico.