A police investigation is under way into a pitch invasion that happened after the Tyrone Senior Football final in Omagh yesterday.
Jubilant crowds raced onto the field at Omagh's Healy Park following Dungannon Clarkes' victory over Trillick in the Tyrone football final.
It has prompted worries about social distancing during the coronavirus pandemic, Ms Foster said.
Dungannon became Tyrone football champions for the first time in 64 years after securing victory in a penalty shoot-out.
The PSNI said it is examining any potential breaches of public health measures at the event.
Chief Superintendent Ryan Henderson said: "We are aware of an incident at a GAA match in Omagh yesterday and will be reviewing all available evidence to determine any potential breach of the Health Protection (Coronavirus Restrictions) Regulations NI 2020 or breach of any other statute identified in respect of any individual."
Earlier Ulster GAA reiterated an appeal for supporters to stay off playing fields following concerns expressed by Northern Ireland First Minister Arlene Foster.
Mrs Foster tweeted: "Significant milestone for Dgn but Covid-19 is no respecter of victories. Other events being responsible. Sport & health will be the losers. @UlsterGAA need to address this. Serious questions arising from videos."
Deeply concerned about the images from yesterday's GAA match. Significant milestone for Dgn but Covid-19 is no respecter of victories. Other events being responsible. Sport & health will be the losers. @UlsterGAA need to address this. Serious questions arising from videos.— Arlene Foster #We’llMeetAgain (@DUPleader) September 21, 2020
In a statement, Ulster GAA said "the on-pitch celebration scenes following the Tyrone County Final in Omagh not only breached GAA protocols but, and much more significantly, public health guidelines.
"Ulster GAA fully realises that this was a moment of great joy for the Dungannon Clarkes Club in winning their first County title since 1956 and congratulations to them on this success.
"The dramatic nature of the conclusion of the game probably added to the exuberance but we are living in a pandemic and the post-match scenes did not portray the Association in a positive light."
They also admitted that it "undoubtedly placed GAA members and their local community at greater risk to Covid-19.
"It also potentially undermines the GAA case, and indeed the case for wider sport, to be permitted to have increased numbers attend our games"
With more county finals due in the coming weeks, Ulster GAA have said similar scenes cannot be repeated or they risk going back to a position where all games will be played behind closed doors.
Ulster also claimed that "the GAA has led the way in its response to this pandemic at community level and has safely returned players and spectators to our games in a safe and responsible way."
Stormont Health minster Robin Swann described the scenes as "disappointing".
However he credited the GAA for developing Covid-19 return-to-play guidance - along with rugby and soccer authorities - and said the events in Omagh were not in keeping with what the organisation was trying to achieve.
"The three sporting codes came up sets of individual guidance and laws and regulations that their own sports and disciplinary procedures could take forward," he said.
"On sight of what we've actually seen and what was actually produced widely across social media I don't think that in any way is in keeping with what those sporting codes and those bodies would expect of some of their players and definitely not of their supporters.
"This virus does not respect any sporting definition, any game, any team. So when I saw that sort of large scale...breaches of social distancing regulations it does concern me and I think...there's no winners when we see that sort of outpouring on social media and across media."