Per Salguero, Sitton asked about playing through the injury before being informed that it could become career-threatening if he didn't repair it.
Veteran Ted Larsen is expected to replace Sitton at left guard for the remainder of the season.
The 31-year-old Larsen is in the midst of his ninth NFL season, and his second with the Dolphins.
Larsen enjoyed stints with the Tampa Bay Buccaneers, Arizona Cardinals and Chicago Bears before starting eight games at left guard for the Dolphins last season.
The 2010 sixth-round draft pick out of NC State has starting experience at both guard positions and center in the NFL.
Overall, he has 73 career NFL starts to his credit.
Sitton signed a two-year deal with Miami during the offseason after spending the previous two campaigns with the Bears.
He played his first eight NFL seasons with the Green Bay Packers before they surprisingly released him prior to the 2016 season.
The 32-year-old Sitton is a four-time Pro Bowler and one-time Super Bowl champion who will leave a big hole in Miami's offensive line despite Larsen's experience.