Television footage showed Uruguay striker Nunez unleashing a flurry of punches at Colombian supporters after jumping into the spectator seating at Charlotte's Bank of America Stadium following an ill-tempered loss to Colombia.
Uruguay central defender Jose Maria Gimenez said players had leapt into the melee out of concern for the safety of family members and loved ones watching the game.
"This is a disaster. Our family was in danger. We had to go to the stands to take out our loved ones, with tiny newborn babies," the Atletico Madrid player said.
"There was not a single police officer...I hope those who are organising this are a little more careful with the families.
"Every game it's been happening because there are people who don't know how to handle a couple of drinks."
The overwhelming majority of the spectators in the crowd were supporting Colombia but there was no segregation between them and their Uruguayan counterparts.
While punches were thrown several Uruguay players climbed into the crowd, seemingly to protect them, with Nunez prominent among them.
The fighting continued for several minutes until police finally intervened.
A group of around 80-100 Uruguayan fans remained in the stadium, on the field, long after the rest of the fans had left.
Ugly scenes also broke out on the field after the final whistle, with players and staff from both teams involved in a mass melee in the centre-circle following Colombia's victory.
