Jese's strike was not enough to give los Blancos victory as Ibai Gomez equalises to ensure Atletico Madrid go clear at the top of La Liga
The Portuguese forward appeared to square up to midfielder Ander Iturraspe, after initially clashing with Carlos Gurpegui, which resulted in the attacker being shown a red card.
Madrid had moved ahead in the 65th minute when Ronaldo set up Jese to score from close range on his first La Liga start.
Athletic equalised through Ibai Gomez after 73 minutes, the substitute finding the net almost immediately after being introduced to the play, and two minutes later Real were down to 10 men after Ronaldo's dismissal.
The result ends a run of five consecutive league wins for Carlo Ancelotti's men and leaves them three points behind Atletico, and level with Barcelona following the latter's defeat to Valencia.
Bilbao maintain their undefeated league record at the new San Mames and have not lost in four league games.
Real Madrid's intentions were clear from the outset and they started at a ferocious pace with Ronaldo and Jese both firing wide within the first three minutes.
However, Athletic soon settled and should have gone in front in the 18th minute when Ander Herrera's superb through-ball found Aritz Aduriz, who escaped the attentions of Pepe before firing wide of Diego Lopez's right-hand post.
Another good chance was spurned by the hosts when Iker Muniain failed to control Oscar de Marcos' fine low cross.
Madrid were subdued but set out to change that immediately after the restart, with Luka Modric unleashing a long-range drive that goalkeeper Gorka Iraizoz did well to tip around the post.
The visitors thought they had taken the lead in the 50th minute when Xabi Alonso's lofted pass was volleyed in by Ronaldo, but the 28-year-old had strayed marginally offside.
Ernesto Valverde's men were indebted to Iraizoz again three minutes later as he repelled Karim Benzema's fierce near-post drive, while Muniain unleashed a curling effort at the other end that went just over.
The Blancos, however, hit the front in the 65th minute when Ronaldo's low cross found the sliding Jese, who prodded in.
The lead didn't last long, as Ibai came on with 16 minutes left and almost immediately drilled in a fine 25-yard effort that went in off the post.
Real were then reduced to 10 men when Ronaldo and Iturraspe clashed and the forward was penalised for raising his hands.
It could have been worse for Real, but Diego Lopez made a superb flying save to tip over Ibai's dipping shot.