Ronaldo was controversially sent off in 75th minute of the 1-1 draw at San Mames following an off-the-ball incident involving Athletic captain Carlos Gurpegi.
The Portugal international 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 confronted by Athletic midfielder Ander Iturraspe, who clearly pushed his head into the Madrid player's face.
Iturraspe ended up with a yellow card while Ronaldo was shown a straight red card - something his team-mates believe was undeserved.
Madrid defender Marcelo said on his club's official website: "It's tough to lose a player in an unfair way. The second-off of Cristiano wasn't fair, although I'm not going to talk about whether the referee had a good match or not.
"We're tired of trying to talk with the referees. They talk with us and we listen to them and it would be good if they also listened to the players."
Midfielder Xabi Alonso added: "It wasn't aggression, the red to Cristiano Ronaldo is excessive. Gurpegi helped a lot and the referee was affected by the reaction of the fans throughout the match. That always happens here, it's nothing new.
"Cristiano shouldn't have done what he did but it is not a sending off, that went against us. We came here with the intention of winning because it was a good opportunity to pick up points in the league but we knew it was going to be a difficult trip".
Ronaldo's red came moments after Athletic had equalised through Ibai Gomez's piledriver after Madrid had taken a 65th-minute lead through Jese.
The result left Madrid in third place in the standings, level on 54 points with defending champions Barcelona who were surprisingly beaten at home by Valencia on Saturday, with new leaders Atletico Madrid now three points clear.
It was undoubtedly a missed opportunity following Barca's unexpected slip-up, but Madrid coach Carlo Ancelotti was nevertheless happy with his side's performance.
He said: "We played well. It was a high intensity match and we hardly let them have any chances. We did especially well in the second half.
"It could have been different because we scored first and dominated the match afterwards. Athletic Bilbao scored a fantastic equaliser, but overall we are going away satisfied with how we played."
On Ronaldo's red card, the Italian added: "It's tough for me to say because I was not close to it, but the pictures show that it was a bit exaggerated.
"It was difficult to try to win the match with 10 players, especially because some of them were tired. I made changes, but not because I wanted to try to win the match, more to put on fresher players to help the team at the end."