The 29-year-old Spain international was angry after Dominic Solanke's goal was allowed to stand, despite the Spurs striker appearing to kick through the back of City centre-back Marc Guehi's leg.
The incident was reviewed by video assistant referee (VAR), which ruled no foul had been committed.
"It was 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," the FA said in a statement.
Rodri, the 2024 Ballon d'Or winner, had voiced his frustration with referee Robert Jones after the match.
"I know we won too much and the people don't want us to win, but the referee has to be neutral," he said. "It's not fair because we work so hard."
Second-placed City trail leaders Arsenal by seven points with a game in hand.