The 21-year-old forward sustained the injury when was hit by Los Angeles Kings defenseman Matt Roy at 9:18 of the first period in the Maple Leafs' 5-0 win on Thursday.
Robertson, who left the game favoring his right shoulder, will attempt to rehab the injury instead of having surgery.
"It's not one of those things where you absolutely have to have surgery," Maple Leafs coach Sheldon Keefe said. "He's going to to rehab it and give it every opportunity to heal appropriately and take it from there, but it's just one of those things that they'll continue to assess it as he moves along."
This is the third significant injury Robertson has sustained in as many seasons.
In his season debut for the Maple Leafs in 2020-21, Robertson sustained a knee injury that caused him to miss four weeks. Then early last season, while playing for Toronto of the American Hockey League, he sustained a a non-displaced fracture in his right fibula that caused him to miss 10 weeks.
"It [stinks] even more because he's been through this before and has had some bad luck with this." Keefe said. "As a young player, it's tough to get traction, but because he has been through it before and each time he's come out a little bit better, that should give him confidence and belief that he's got reasons to continue to work and push through and as a result, gives us that same confidence."
Robertson, who was selected by Toronto in the second round (No. 53) of the 2019 NHL Draft, has five points (two goals, three assists) in 15 games this season and seven points (three goals, four assists) in 31 NHL games.
