The 24-year-old defender hit out at Mourinho after Barcelona's ill-tempered 3-2 second-leg win over Real in the Spanish Super Cup on Wednesday night.
Mourinho could face action after pictures showed the Portuguese manager making contact with the face of Barcelona's assistant manager Tito Vilanova.
"Someone has to take action on the matter," said Pique.
"Mourinho is destroying Spanish football. There is talk about the Catalans, but the problem is with Madrid.
"I think it's going too far. It cannot always end well. In the end this will end very badly."