Miami placed Sitton on injured reserve on Friday afternoon.
The four-time Pro Bowl selection signed a two-year contract with the Dolphins this offseason in free agency.
The Dolphins had Sitton listed as their starting left guard on their Week 2 depth chart, ahead of Ted Larsen. Jesse Davis was the only other guard listed on the depth chart and he is set to start at right guard.
Sitton missed practice on Wednesday and Thursday.
"If there was something where we weren't going to have Josh or any of our guys, we'll look at what the other guys do well and try to put them in situations to be successful," Dolphins offensive coordinator Dowell Loggains told reporters Thursday.
"Obviously, New York has good inside players. And some weeks the matchup would be different, if it was a tackle or a center, however it is. Does it change things? Absolutely. If you didn't have someone at any position, you look to [say], 'How is this matchup going to affect us in the game?'"
Sitton, 32, was a fourth-round pick by the Green Bay Packers in the 2008 NFL Draft. The 6-foot-3, 318-pound offensive lineman played his first eight seasons with the Packers, winning a Super Bowl in 2011.
The Dolphins face the New York Jets in an AFC East matchup at 1 p.m. on Sunday at MetLife Stadium in East Rutherford, N.J.