The passenger, a 25-year-old man, was arrested after he and his girlfriend fled the scene, about 50 miles west on Phoenix on Interstate 10, KNXV-TV, Phoenix, reported.
The couple was arrested when they returned to the scene about 30 minutes later.
The suspect, Maquel Donyel Morris, yelled, "I'm gonna flip this bus," as he attacked the driver, police said.
Harquahala Fire District Chief Dan Caudle said three seriously injured passengers were airlifted to Phoenix hospitals. Twenty others were treated at the scene and 12 were not hurt, KPHO-TV, Phoenix, reported.
The bus was eastbound when the suspect allegedly began beating the driver about 1:45 a.m. The bus crossed into the median, where it stopped.
The driver did a "good job of maintaining control," Caudle said.
Passenger Gregory Fort, a commercial truck driver from Oakland, Calif., said the driver did a "tremendous job."
"The guy was beating him uncontrollably, hitting him in the face, head," Fort said. "There is nothing anyone could have done better. He deserves a [commendation] for handling bus the way he did."
The suspect's girlfriend told police he was "paranoid" and "did not want to be on the bus," KTVK-TV, Phoenix, reported.
Morris faces 48 counts of felony endangerment, 24 counts of assault and three counts of aggravated assault.
Greyhound said it would send two buses to pick up uninjured passengers and take them to their destination.