The Kerr effect in an optical passive ring-resonator gyro (OPRG) is analyzed considering the optical source coherence. This effect causes the drift to be much greater than the theoretical limit of rotation sensing given by the detector shot noise for an OPRG with high sensitivity. The drift depends not only on the fluctuation of the power splitting ratio between two counter-propagating beams in the resonator, but also on the frequency modulation indices used to set the operating point. Square-wave intensity modulation of the incident beam is also discussed as a way to eliminate the drift. It is found that the ratio of the signal-beam round-trip time in the resonator to the modulation period must be an integer to optimize the drift elimination. From the practical viewpoint, the requirement for the modulation parameters is analyzed numerically.