Indianapolis also signed defensive tackle Johnathan Calvin to the practice squad and placed cornerback Rashaan Melvin on injured reserve.
The Colts announced the moves on Wednesday.
Kaaya, 22, was a sixth round pick by the Detroit Lions in the 2017 NFL Draft. The 6-foot-4, 325-pound quarterback participated in the Lions' 2017 offseason program and training camp but was waived on Sept. 2. The Carolina Panthers claimed Kaaya off waivers the next day. He spent the last nine weeks on the Lions' practice squad and spent the first six weeks of the season on the Panthers' active roster. Kaaya was on the Lions' active roster in Week 7. He has never appeared in a regular season game.
The former Miami Hurricanes quarterback completed 24-of-40 passes for 291 yards, three scores and one interception in four games this preseason. He also lost a fumble.
Kaaya completed 720-of-1,188 passes for 9,968 yards, 69 touchdowns and 24 interceptions in 38 starts for the Hurricanes.
Calvin entered the NFL on May 5 as an undrafted free agent signing by the Green Bay Packers. He was waived during final roster cuts on Sept. 2 and signed to the Colts' practice squad on Sept. 12. He was waived two weeks later.
Melvin has 116 tackles, 21 passes defensed, three forced fumbles, three interceptions and two special teams tackles in 37 games for the Colts, New England Patriots and the Baltimore Ravens. He had 36 tackles, 13 passes defensed and three interceptions in 10 starts this year for the Colts. Melvin's season ends due to a dislocated bone in his hand.