The 21-year-old sustained the damage during his side's 25-18 home defeat by Leigh Centurions in the Super 8s Qualifiers a fortnight ago.
However he played on through the pain barrier in the second half without realising the extent of the injury.
Scans have since uncovered the fracture and Lawler will miss the rest of the season as Rovers seek to preserve their Super League status, although he is expected to make a full recovery in time for the start of pre-season training in November.
"We're disappointed to be without George for the next few weeks but he needs to be commended for his courage and bravery in playing through such an injury. "He really is as tough as they come," said the club's heady of rugby Jamie Peacock.
"It was only after the game that scans picked up the fracture and unfortunately it means he won't play again this season.
"However, the specialists and our own medical staff are very confident there will be no lasting damage and we'll have him back for the start of pre-season training."
