Letterkenny club felt they would have been granted a replay after Termon's injury-time equaliser was scored with 16 men on the field, but have opted not to pursue the issue.
ST Eunan's have decided not to take an appeal against their Donegal SFC quarter-final defeat by Termon on Saturday, saying that to do so would "risk undermining [the competition's] integrity".
Their case centred on Termon's equalising point that sent the game to extra-time, fisted over by substitute Enda McCormick in stoppage time.
Termon had just introduced McCormick but the player he was replacing, John James Sweeney, had not made his way off the pitch and was visibly engaging a number of St Eunan's defenders during the play that led to the equaliser.
In a statement released following two days of discussion between the team and the club's executive committee, St Eunan's say they are "confident a replay would have been granted" had they pursued an appeal.
But citing the "disruption" it would cause to the championship and how it would undermine the competition's integrity, the Letterkenny club have opted against pursuing the issue through the boardroom.
"Following a meeting of the Executive Committee this afternoon, St Eunan's GAA can confirm that we will not be appealing the outcome of last weekend's championship fixture against Termon.
"The club believes it has a strong case under the rules, given that the winning score resulted directly from a 16th player, while the player being replaced was directly involved in engaging our defenders at the time. St Eunan's players consulted the match referee at the time and asked him to consult with the linesmen and fourth official which he declined to do.
"On this basis, had an appeal been lodged, we are confident that a replay would likely have been granted.
"However, St Eunan's is equally mindful of the principles of sporting integrity and the fact that the incident was not the result of any deliberate act on Termon's behalf. Our disappointment lies with how the situation was handled by the match officials. The game should not have been allowed to restart until the Termon player being substituted had fully exited the field of play. It was not Termon's fault, nor was it ours, and we strongly believe that standards of communication and officiating must improve to prevent such incidents in future.
"Ultimately, while we remain convinced that our case had merit, the Executive Committee has decided that pursuing the matter through appeal is not in the best interests of the competition. An appeal would cause disruption to the championship and risk undermining its integrity.
"For these reasons, St Eunan's will not be taking the matter further and wish Termon the very best in the remainder of the championship," the club's statement read.