Rodgers called the performance of the officials "horrendous" in the wake of his side's 2-1 defeat by Manchester City and also questioned the wisdom of allowing Bolton-based referee Lee Mason to take charge of a game involving a Manchester club.
He later denied disputing the integrity of the officials, arguing he was "only speaking honestly after the game" and decided not to contest the FA charge.
An FA statement read: "Rodgers was charged for a breach of FA Rule E3 in that his post-match comments following Liverpool's game at Manchester City on 26 December 2013 constituted improper conduct in that they called into question the integrity of the match referee, and/or implied that the match referee was motivated by bias; and/or brought the game into disrepute; and/or amounted to a failure to act in the best interests of the game."
The Irishman had been infuriated by an incorrect offside decision against Raheem Sterling with the City game still goalless, and he also felt Luis Suarez should have been awarded a second-half penalty.
Explaining his post-match comments the day after the defeat, Rodgers said: "I certainly wasn't questioning the integrity of referees.
"(The comment) was more than logical in terms of having a referee from that part of the world refereeing a game in Manchester.
"I wouldn't suspect that Mike Dean, from the Wirral, has refereed many games for Liverpool over the years.
"I was only speaking honestly after the game. I think the FA and the Referees' Association know perfectly well I am someone who engages in supporting the referees."