Rourke left the game in the second quarter and was examined in the blue medical tent before heading to the locker room for further evaluation.
The 49ers said their backup quarterback, who started Thursday night's game, suffered a rib injury.
While initial X-rays did not reveal a fracture, Rourke was having trouble breathing, prompting the 49ers to send him to the hospital.
"He was struggling to breathe, so they took him to the hospital just to check on everything, make sure it's all right," head coach Kyle Shanahan said after the game. "X-rays in here didn't show it was broken or anything, but we'll see what we find out tomorrow."
The 49ers said Rourke was released from the hospital Thursday night. His release, combined with his presence at the team facility Friday, is an encouraging sign. However, the team has not provided an official update on his status.
Adrian Martinez replaced Rourke against the Titans and finished the game.
Shanahan believes Rourke sustained the injury on a goal-line draw play in the second quarter. The decision to send him to the hospital was made out of an abundance of caution.
"When you don't see something on an X-ray, and someone's struggling with something, then you just got to go cover your bases and look for more," Shanahan said.
Rourke completed 12 of 14 passes for 101 yards against the Titans. He also rushed for 14 yards on three carries. He is competing with Martinez for the QB3 job behind Brock Purdy and Mac Jones.
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