Valverde was dismissed by the Catalan club on Monday, and Zidane conceded that a "crisis" is never far away at Spain's two biggest teams.
Asked for his opinion on the decision, Zidane said: "I'm not going to get involved with that. First of all, I'm sorry for him. He's a coach who has shown he is very good and I have respect for him.
"At big clubs, that's how things are and nothing will change. I know that if we lose two games, I will be criticised in the same way as I was a month and a half ago. We can't always play well."
And he told the media: "We have to do all that we can and then it's you to speak well or badly [of us]."
Madrid face Barca again on March 1 at the Santiago Bernabeu in LaLiga and in that match, Zidane will be up against former Real Betis coach Quique Setien after the 61-year-old replaced Valverde this week.
"I expect a Barcelona team that will compete as they have always done with all their coaches," the Frenchman said.
"He will do his job and we will do ours. The rivalry will always be there. I expect a competitive team."