Gallagher suffered the injury during the second period of Tuesday's contest against the Carolina Hurricanes after falling and colliding headfirst with teammate Ben Chiarot's knee.
The 27-year-old winger did not come out for the third period.
Gallagher ranks second on the club in both goals (15) and points (32) through 40 contests this season. He hasn't missed a game since the 2016-17 campaign.
The Canadiens have lost three straight games and sit four points back of Thursday night's opponent, the Tampa Bay Lightning, for third place in the Atlantic Division.