Newcastle United's Isaac Hayden has been charged with improper conduct by the Football Association after the defeat at Chelsea on 13 March.
"Some performance against 12 men today," Hayden posted - comments deemed in breach of FA Rule E3.1 for attacking the integrity of the match official.
Chelsea won thanks to a late goal from Kai Havertz, who had earlier escaped a red card for an elbow on Dan Burn.
Newcastle were also incensed by Coote's failure to award them a penalty when Jacob Murphy went down in the box.
Hayden did not feature in the match, after undergoing surgery on a knee injury in February.