Rodgers was incensed by two decisions that went against his side in December's 2-1 defeat against Manchester City.
The Liverpool boss was first angered when Raheem Sterling's effort was disallowed for offside, and further irked when Luis Suarez was denied what the Northern Irishman felt was a penalty after a challenge from Joleon Lescott.
After the defeat, Rodgers described the official's performance as "horrendous", adding: "Hopefully we won't have a Greater Manchester referee with Liverpool-Manchester games in future."
The FA originally stated that the comments "called into question the integrity of the match referee, and/or implied that the match referee was motivated by bias".
While Rodgers later said he was "talking in terms of geography" rather than "questioning the integrity of referees", the FA confirmed on Wednesday that the 40-year-old had accepted the charge.
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