In a statement released to Independent.ie a garda spokesperson said: "No complaint has been made to An Garda Síochána, at this time."
The melee broke out at the end of full-time at the Cusack stand side of the pitch as both teams made their way to the dressing rooms at the end of full-time.
Footage on RTÉ's coverage appeared to show that starting players, substitutes and a number of extended panel members, who were not part of the match day squads, were involved in the incident.
The footage also showed evidence of an alleged eye-gouging incident which has become one of the main talking points from the game.
When asked if Gardaí can investigate an incident without a complaint being made, the Garda spokesperson said: "Enquiries into a potentially criminal incident can begin before an official complaint has been received by an injured party."
Meanwhile, former GAA president Liam O’Neill believes Croke Park must “get to grips” with the issue of mass brawls after yesterday’s chaotic scenes between warring Armagh and Galway players at the end of normal time.
The Laois official called for extended panellists who are not part of match-day squads to be kept in the stands. He believes the GAA must ensure, where possible, that two teams don’t head for the same tunnel at half-time or full-time at a time when tensions are often inflamed.
And he also called on the Association to “tidy up our act” after several controversies this year where proposed suspensions arising from clear bouts of indiscipline have been quashed on technicalities.
Speaking on RTÉ’s Morning Ireland, O’Neill expressed disappointment at yesterday’s turn of events during the All-Ireland SFC quarter-final.
“We’d an absolutely fantastic game yesterday in Croke Park, we had everything we want in Gaelic games, and yet this morning you’re talking to me about violence and that’s a huge pity,” he said.
On the question of two teams heading for the same tunnel, with incendiary consequences, O’Neill recalled: “In my time as GAA president I made a huge effort to clear the sidelines, I had it down to one manager and one runner on the sideline. I think that’s more than enough.
“I really think we have to get to grips with this situation. We’d a situation in Croke Park yesterday where we had excellent dressing-rooms on both sides of the field; we could have used one for either team in the first game, one for either team in the second game, and avoided this.
“And people say, around the country, it’s the same – it’s not. In Portlaoise we have two dressing-rooms; we have two ends, the players come out of either end of the same stand, and there’s never a difficulty. Admittedly, it was in response to a situation which developed – but we solved it, and I think Croke Park really have to get to grips with this.
“There’s no need for the extended panellists to go into the dressing-rooms at half-time. What are they contributing there anyway?
“So, we really should have people sitting in a stand and have the manager and his selectors in the dressing-room with their players. It would be much tidier, and we wouldn’t have had this situation we had yesterday.”
On the more general subject of the GAA’s creaking disciplinary system, O’Neill accepted there is an issue.
“That has to be admitted, that we have to tidy up our act,” he said. “Really, we should move to a situation where penalties are imposed on the day. We had that at one stage in the GAA where we cleared people from the pitch for incidents with a black card that lasted for the entire game. That meant that people just weren’t doing the sort of things they’re doing now. We have to get back to that.
“We can’t legislate for everything that happens. We can’t anticipate in advance. Who could have thought that would have happened yesterday?
“But I think the responsibility on sports administrators is how we handle situations when they develop. And it’s up to us now to address what happened yesterday and make sure it simply does not happen again.”
O’Neill also addressed another hot potato – rising complaints about the GAA’s new split-season – as he reflected on the need to finish yesterday’s quarter-final with a penalty shootout instead of finding room in the squeezed calendar for a replay.
“I don’t like it (penalties), I have to say. But, do you know what, we legislated for that. People got into a rule, at some stage, and voted on that and we have to accept, if it’s in the rules, that’s what happens,” said veteran official, who served as GAA president from 2012 to 2015.
“But it’s not our way of doing things. We were frog-marched into this thing of the split season, getting the championship over early, by the media and by people saying that we have to look after the clubs … it hasn’t really worked.
“Yeah, we were kind of told this was the way it should be and we responded and thought it was good to go along with the general narrative that it was better. It’s not, quite frankly.
“We’ve given away September, given away August, to other sports and I thought we had a very good system – and somehow we changed it and it hasn’t worked, and we have to nip that now and go back and say this hasn’t worked, let’s do what we always did.”