Bailly limped out of the 1-1 draw at Old Trafford after he collided with Mateo Kovacic during the second half of the Premier League clash.
Scans on Monday confirmed damage to the medial ligament in his right knee, meaning he will miss United's last two games of the season and the Africa Cup of Nations this summer.
"I would like to thank everyone for the messages of love and support I have received," the Ivorian wrote alongside a picture of himself in a leg brace with his thumbs up.
"It's tough, but one must accept that some injuries are unavoidable. It's true that this season hasn't smiled down upon me very much, but I will return soon and with more strength next season, God willing."