Stewart announced the decision in a statement Thursday. His plans for the rest of the season have not been determined.
Jeff Burton will drive Stewart's No. 14 Chevrolet in Michigan.
Stewart also sat out Sunday at Watkins Glen, a day after he struck and killed Kevin Ward Jr. during a sprint car event in Canandaigua, New York. After Stewart clipped Ward's car, sending it spinning, Ward got out of the car during the caution period, walked down the track and was hit by Stewart.
Ward's funeral was Thursday in Boonville, New York. He died of blunt force trauma.
Stewart's dirt-racing career is on hold, and he could face criminal charges.
Stewart has not commented since he made a statement Sunday.
"There aren't words to describe the sadness I feel about the accident that took the life of Kevin Ward Jr.," Stewart said in the statement.
The three-time NASCAR champion is winless and 21st in the Cup standings this season. He would have needed a victory in one of the final four races to have a chance to become eligible for the Chase for the Sprint Cup championship.
It's not clear what sitting out does for Stewart's Chase chances. NASCAR rules says a driver must either qualify the car or race the car each weekend. NASCAR does have the power to grant a waiver.
Stewart has 48 career Cup wins in 542 starts.
Ontario County Sheriff Philip Povero said that investigators don't have any evidence at this point to support criminal intent. The investigation could last another two weeks.
His sponsors have stayed with him as the investigation unfolds. Mobil 1 has served as the primary sponsor of the Stewart-Haas Racing No. 14 Chevrolet for 19 of the 22 Sprint Cup Series races this season.
"The investigation continues, and we are monitoring the situation," ExxonMobil spokesman Christian Flathman said Thursday.
Regan Smith started for Stewart at Watkins Glen and finished 37th.
Burton made only two starts this season, the last coming in July at New Hampshire. The 21-time Sprint Cup winner parted ways with Richard Childress Racing at the end of the season, and Ryan Newman replaced him at RCR. Burton works as an analyst for NBC.