Barca and Real players confronted each other on the pitch during their explosive Spanish Super Cup second leg at the Nou Camp where red cards were flashed and Mourinho jabbed Barca assistant coach Tito Vilanova in the eye.
"We have decided not to [make an official complaint against Mourinho]" Barca president Sandro Rosell on Monday told a meeting of supporters groups, who had called for the club to act.
"The public have give their opinion. They are very clear that what this person did wasn't right. The [Spanish football] federation will act if they want to.
"We have to act sensibly, reduce the levels of tension and try not to stir it up or we'll end up killing each other in the street. The problem isn't ours, it is theirs."
Last season's four 'clasicos' in under a month were characterised by an increasingly poisonous atmosphere between the sides, which reappeared in the second half of last Wednesday's match.
Marcelo's lunging tackle on Barca newboy Cesc Fabregas sparked a melee on the touchline and led to red cards being shown to the Brazilian, his German team-mate Mesut Ozil and Barca striker David Villa.
The most controversial incident, however, was when Mourinho walked up behind Vilanova in the middle of the fracas and poked him in the eye, before strolling away as the Barca man cuffed him on the back of the head.
ANGRY EXCHANGES
Mourinho denied the incident had happened after the match and his spokesman Eladio Parames told daily El Mundo: "Jose doesn't ask for forgiveness.
"He is very clear that he is defending Madrid. All he is interested in is his work and to make sure that the club where he works wins as many titles as possible.
"But if his rivals work with the dark arts, he is absolutely clear he will not remain impassive."
The angry exchanges between the Spanish internationals in the two sides have also prompted speculation that the bad blood between Real and Barca will disrupt the harmony of Spain's World Cup-winning squad.
Two of Spain's senior figures, Real captain Iker Casillas and Barca midfielder Xavi could be seen having a heated exchange on the pitch after Wednesday's clashes.
"The situation in the national team is at its limit," Spain and Malaga midfielder Santi Cazorla was quoted as telling regional newspaper La Nueva Espana.
"As hard as they try to avoid it, these things influence the relationships. We can't do anything except wait for them to sort it out as quickly as possible, to reach an agreement, which is what we all want.
"If they keep on having these problems it could start to have an influence and in the end this could be prejudicial for the group."
