Celtic were on Friday handed a lifeline in Europe's premier club competition after Legia were punished for fielding an ineligible player in their 6-1 aggregate win over the Scottish champions in the third qualifying round.
Legia boss Henning Berg swiftly vowed that his club would appeal, and Mioduski on Sunday issued an open letter stating his side's case and asking the Hoops to "settle this matter honourably".
After reflecting on Celtic's "beautiful" history, Mioduski said: "I therefore call on you to refer your best traditions of honour and honesty, that your famous club has been know for during the last 126 years. Do not destroy the beautiful legacy, that you inherited from the past generations of 'The Bhoys'.
"I call on you to act according to the spirit of the game and rules of fair play - to issue a joint statement to the UEFA disciplinary bodies. Let's meet in Warsaw or Glasgow and settle this matter honourably."
Bartosz Bereszynski, who came on as an 86th-minute substitute as Legia won the second leg 2-0 at Murrayfield on Wednesday night, was handed a three-match ban after a red card in the Europa League last season.
Legia believed he was free to play after missing the two games against St Patrick's Athletic as well as the first leg against Celtic in Warsaw.
However, the player was not included on Legia's squad list submitted to UEFA for the St Patrick's games and thus his suspension was not deemed to be served.
Celtic were awarded a 3-0 win, meaning they went through on away goals after a 4-4 draw and were drawn against Slovenian side Maribor in the Champions League play-off, with the Warsaw club dropping into the Europa League.
Mioduski described UEFA's punishment as "deeply unfair and contradictory to the fair play rules", and referenced Celtic's "rich heritage" and 1967 European Cup triumph in attempting to persuade them to agree to a rematch.
"We found it deeply unfair and contradictory to the fair play rules to see our dreams shattered by the last four minutes of the already decided second leg match, when Bartosz Bereszynski was brought on the pitch," Mioduski wrote.
"Our player had served the full sentence and intentionally missed the first three matches in the Champions League qualifiers this season."
He continued: "Imagine Jock Stein and Billy McNeill deprived of the chance to achieve the biggest triumph in their career by an application form, filled improperly by a club employee acting in good faith.
"Would any true legend of Celtic FC accept the qualification won despite the double defeat suffered on the pitch? I am strongly convinced that their legendary Celtic pride would not allow them to do that.
"How could they then look in the eyes of their humiliated fans?
"Willie Maley, the legendary manager of Celtic FC, once said that in your stadium 'a man is judged by his football alone'. Only you can decide whether this noble credo will be replaced by an opportunistic use of legal loopholes."
In response to statements made by Legia Warsaw today, a spokesperson for Celtic Football Club said: "We are disappointed by Legia Warsaw's comments.
"This is entirely a matter for UEFA and its processes. Accordingly, we will reserve further comment for the appropriate time."