A rotor system with double time delays supported by the high-speed self-acting gas-lubricated bearings with three-axial grooves is modeled to implement active delay control of the system. The differential transformation method is employed to solve the time-dependent compressible gas Reynolds equation due to its rapid convergence rate and minimal calculation error. Based on the precise integration method, a calculation method is proposed to analyze the dynamic responses of a gas bearing-rotor nonlinear system with time delays. The motion analysis of the self-acting gas-lubricated bearing-rotor system with double time delays is implemented by the orbit diagrams, the time series, and the phase diagrams. The influence of time delays and feedback control gains on the dynamic responses of the bearing-rotor nonlinear system is analyzed. The numerical results show that the amplitude of the responses of the system with time delays control is reduced, the motion is more stable and good control effect is achieved when the chosen feedback control gains match the time delays of the bearing-rotor system.