"Uganda is not a monarchy where leadership is passed on from father to son," he was quoted by The Ugandan Daily Monitor as saying.
The 39-year-old military leader is a one-star general in control of a 10,000-unit Special Forces unit. The general said in a statement published by the newspaper the power to appoint government officials lies with the Ugandan people.
Ugandan law prevents military figures from holding political office.
The offices of the Ugandan newspaper were raided by military forces last month. Detectives forced at least two radio stations off the air and seized the newspaper's printing presses May 20 because of a letter involving Kainerugaba's political trajectory.
The Daily Monitor published a letter from a top military general that asked for an inquiry into reports that orders were given to kill anyone who stated opposition to Kainerugaba.
Parent company Monitor Publications Ltd. said its editorial policy had been violated and steps were taken to ensure extra attention was given to stories that may have an impact on national security.