Stewart is expected to return in time for preseason testing in January 2014.
Martin was granted his release Monday from Michael Waltrip Racing to join SHS. He will drive the No. 14 this weekend at Bristol and in 11 other races.
Dillon will race Oct. 20 at Talladega. Dillon filled in for Stewart on Sunday at Michigan and finished 14th.
Stewart broke his right leg in a sprint car crash Aug. 5 at Southern Iowa Speedway.
"Obviously, I'm disappointed to be out," Stewart said in a statement. "But the team is in very good hands with Mark Martin and Austin Dillon. Mark is someone I've looked up to my entire career and I have a tremendous amount of respect for him. Austin is a great young talent, and he showed that Sunday at Michigan."
Greg Zipadelli, competition director at Stewart-Haas Racing, said Stewart needed to focus on his health and be ready to return next season.
Brian Vickers will drive the No. 55 Toyota for MWR in 12 of the final 13 races. He was originally scheduled to drive just three more times. Michael Waltrip will race the No. 55 at Talladega as previously scheduled.
Waltrip said Martin achieved all of the goals both he and the team set when he began a 24-race schedule in the No. 55 in 2012.
"With Mark, we accomplished everything we hoped to when he came to this organization, which was bringing that winning confidence to the team, the experience to help elevate our program and ultimately find the full-time driver for the future who can race for a NASCAR Sprint Cup championship," said Waltrip, a two-time Daytona 500 champion.
Martin has 40 wins and 56 poles in 870 career Sprint Cup starts.
MWR announced last week that Vickers will drive the No. 55 Toyota for a full Sprint Cup schedule beginning next season.
"No one wants to see Tony out of the 14, but I am appreciative of the opportunity to get more seat time in the 55 as a result," Vickers said.
Martin will be the third driver in the No. 14 this season. Max Papis drove Toyota in Stewart's absence at Watkins Glen.
The No. 14 car is 13th in the owner standings.
"For the No. 14 team, our focus is on positioning them for a run at the owner championship," Zipadelli said. "Mark Martin and Austin Dillon give us the best opportunity to do that, and we certainly appreciate Michael Waltrip Racing and Richard Childress Racing for making Mark and Austin available to us."