Watson injured his right hand in the first quarter of Saturday's game against Louisville and did not return.
Head coach Dabo Swinney told ESPN during his halftime interview that Watson broke his hand.
Watson was replaced by senior Cole Stoudt.
Entering Saturday, Watson had completed 73 of his 106 pass attempts for 1,181 yards and 12 touchdowns with one interception this season while adding 137 yards and three touchdowns on the ground.
Clemson defeated Louisville to improve to 4-2.