An acoustic ring resonator employing a two-dimensional surface phononic crystal is proposed for high-sensitivity detection in binary gas mixtures. Band analyses and frequency-domain simulations via the finite-element method reveal that a single band for spoof surface acoustic waves appears at ultrasonic frequencies around 58 kHz where modification of its dispersion due to varying gas composition results in a linear shift of the resonance frequency. The shift rate is -17.3 and 8.8 mHz/ppm for CO2 and CH4, respectively. The linear shift of resonance frequency is experimentally validated. In addition, the ring resonator can also be employed to track acoustic intensity variation with gas concentration, where exponentially decaying intensity for low concentrations leverages high-sensitivity operation.