Resonator micro optic gyro (RMOG) with a waveguide-type ring resonator is a promising candidate for applications requiring small, light and robust gyros. In optical passive ring resonator gyros, clockwise (CW) and counter clockwise (CCW) lightwaves are phase-modulated at different frequencies to reduce the backscattering induced noise. The effectiveness of this technique, however, is determined by the carrier suppression level. In this paper, the influence of the carrier suppression level on the gyro performance is examined experimentally for an RMOG system with a silica waveguide resonator. In our experiment, carrier suppression is applied onto both the CW and the CCW lightwaves at the same time to achieve higher total suppression. We show that carrier suppression as high as 100 dB can be achieved by optimizing the amplitude of the phase modulation. A bias stability of 0.46 /s in 50 seconds is demonstrated in an RMOG with a silica waveguide ring resonator having a ring length of 7.9 cm. This is the best result reported to date, to the best of our knowledge, for waveguide-type ring resonator gyros of this size.