Mickey Harte and Joe Kernan have spoken out against the proposal to segregate fans at GAA matches.
The brawl between Derrytresk and Dromid players last weekend resulted in Dr Crokes proposing that supporters at next month's All-Ireland club semi-final against Crossmaglen to be segregated.
Kernan believes segregation would be the death nail for the GAA, while Harte has branded the suggestion as outlandish.
"The day segregation comes into the GAA is the day you shut the gates. Crowds being able to mingle is what the Association is built on," Kernan told the Irish Examiner.
"People from England come to watch our games and they admire supporters being able to sit with one another.
"Unfortunately, there have been incidents of late - and I'm sure there have been incidents in Kerry as well as everywhere else - that are regrettable.
"It's certainly not the way we have been brought up. But if we go down the road of segregation the GAA might as well shut up shop."
Speaking to the Irish News, Harte added: "I think this is a serious over-reaction. No-one condones violence but we also have to keep it in context and very often it does appear worse than it actually is.
"Sometimes pictures tell a story that over-exaggerates and, when you go into the finer detail, it may not have been quite as drastic as someone had reported it.
"But to think someone would take that, or an incident of this nature, and ask for something as drastic as segregation is outlandish.
"The very thing that makes us stand apart from other organisations is the ability of our supporters to mingle with each other. It would be a serious, serious detrimental step."