Coote was sacked last December by Professional Game Match Officials Limited (PGMOL) after a video emerged on social media of him making derogatory remarks about Klopp in July 2020.
One of the comments referenced the German's nationality, and the FA has now charged Coote for an "aggravated breach" of its rules.
The FA added that the 42-year-old would face no further action concerning separate allegations of gambling misconduct, which he had always denied.
Coote was banned by UEFA in February from officiating in European competitions until June 30th, 2026, after another video emerged of him snorting a white powder while in Germany for Euro 2024.
In January, Coote came out as gay in an interview with The Sun newspaper, having hidden his sexuality throughout his career in refereeing.