Mourinho, 60, was shown a yellow card by English referee Anthony Taylor during the bad-tempered final in Budapest on Wednesday, which Sevilla won on penalties after a 1-1 draw.
He criticised Taylor in his post-match news conference, and was later filmed in the stadium car park calling him a "disgrace" in a profanity-laced tirade.
Both clubs also face charges over the conduct of their supporters and players.
They have both been charged with throwing objects, lighting fireworks and improper conduct of their team.
UEFA said it "vehemently condemns the violent behaviour" towards Taylor and his family and intends to "carefully assess" these incidents to strengthen the security of its officials in the future "in close collaboration with local police and airport security," a spokesperson told AFP.
"Such actions are unacceptable and undermine the spirit of fair play and respect that UEFA upholds," the spokesperson said.
"Referees play a crucial role in ensuring the integrity and fairness of the game, and their safety and well-being are of utmost importance."
Sevilla have an additional charge for invasion of the field of play, while Roma have also been charged with acts of damage and crowd disturbances.
Taylor booked 13 players, seven of them from Roma, while the bad-tempered game was littered with delays due to the unusually high number of fouls.
The English referee was then targeted by angry Roma supporters at Budapest Airport.
Taylor and his family were eventually led out of a cafe and into a secure area by airport security.
An Italian man was charged with affray after the ugly clash.
Referees' body PGMOL said: "We are appalled at the unjustified and abhorrent abuse directed at Anthony and his family as he tries to make his way home from refereeing the UEFA Europa League final."
It was Sevilla's seventh Europa League triumph and the first European final defeat of Mourinho's managerial career.