Rodri was incensed with referee Robert Jones' decision to allow Dominic Solanke's goal in City's 2-2 draw at Tottenham in the Premier League on 1 February.
He felt Solanke's goal should have been ruled out because the Tottenham striker kicked the back of City defender Marc Guehi's leg in the act of scoring.
"I know we won too much, and the people don't want us to win, but the referee has to be neutral," Rodri told Australian broadcaster Stan Sport after Tottenham fought back from two goals down.
"It's not fair because we work so hard. When everything is finished, you are frustrated."
The FA announced on Friday that the 29-year-old had been charged with misconduct over his remarks.
"It's alleged that the midfielder acted in an improper manner during a post-match media interview by making comments that imply bias and/or question the integrity of a match official and/or match officials, contrary to FA Rule 3.1," an FA statement said.
Rodri has until February 18 to respond to the charge, with the potential for a suspension if he is found guilty.
That would be a major blow to City's hopes of overhauling the four-point gap to Premier League leaders Arsenal.
In 2023, then Liverpool manager Jurgen Klopp was handed a touchline ban after suggesting referee Paul Tierney had "something against" his side.