MacArthur, 28, sustained the injury shortly after kick-off in Saturday's defeat by Northampton Saints in the European Champions Cup.
The Scotland cap is expected to be discharged from hospital on Tuesday.
"We look forward to him re-joining the squad when he has fully recovered," said Warriors head coach Gregor Townsend.
"Pat has featured regularly for us this season and has been in good form, while also stepping up as a leader, so it's disappointing to lose him for the next few weeks."
Glasgow Warriors hooker Pat MacArthur faces up to six weeks out after having surgery on a jaw fracture.