Stewart's decision was announced Wednesday and Jeff Burton will once again replace him in the No. 14 Chevrolet. Burton also drove Stewart's car at Michigan last weekend.
Stewart-Haas Racing is allowing Stewart to take the time he needs away from the track, and the team statement indicated the three-time NASCAR champion is taking it week-by-week. Stewart will not have to decide on another race until the Aug. 31 event at Atlanta.
Stewart has been in seclusion since the Aug. 9 incident at an upstate New York dirt track. He has not commented since he issued a statement the day after Ward's death.
Stewart also pulled out of the race at Watkins Glen, which began about 12 hours after his car hit Ward at event in Canandaigua, New York. Ward's car had spun into a wall as he and Stewart raced for position, and the 20-year-old climbed out to confront Stewart.
Authorities said the first car to pass Ward had to swerve to miss hitting the driver, who was gesturing toward the NASCAR star. The front of Stewart's car then appeared to clear Ward, but Ward was struck by the right rear tire and hurtled through the air.
Ward died of blunt force trauma, and authorities are still investigating the incident. Ontario County Sheriff Philip Povero has said investigators did not have any evidence to support criminal intent by Stewart, but the investigation is expected to last at least another week. Povero said this week were no new updates.
None of Stewart's peers has spoken to the driver since the accident. But they've come to his defense, with six-time champion Jimmie Johnson saying nothing he's seen has convinced him that Ward's death was anything but a tragic accident.
Kevin Harvick, a longtime friend of Stewart's and his teammate at SHR, also bristled at the negative portrayal of Stewart since Ward's death.
"I've known Tony Stewart for a long time ... I know for sure that Tony Stewart is not going to run over somebody," Harvick said. "I don't think there's anybody in this garage that would. It would be hard to find somebody in the racing world that could point that car, just run somebody over."
NASCAR last week announced a rule that prohibits drivers from exiting from a crashed or disabled vehicle -- unless it is on fire -- until safety personnel arrive.
Stewart has 48 career Cup wins in 542 starts. He has one career win at Bristol, in 2001.
Burton made just his third start of the season last week. The 21-time Sprint Cup winner parted ways with Richard Childress Racing at the end of the season and is transitioning into an analyst for NBC. Burton won at Bristol in 2008 and has 15 top-10 finishes in 40 career starts.