The result kept Bilbao's unbeaten home record in La Liga intact, but Valverde has admitted that he had hoped for all three points.
"I thought our first half was very good. And if I'm honest, I was hoping for a win after all the good work we'd done," Valverde told reporters at a press conference.
"It's only normal, though, that Real Madrid would improve in the second half because they are a team full of champions."
Cristiano Ronaldo was dismissed for Los Blancos during the ill-tempered affair.