In underground caverns excavated for compressed air energy storage (CAES), the surrounding rock serves as the primary carrier of high-pressure gas. Under high internal pressure, crack propagation in the surrounding rock can compromise the tightness and structural integrity of underground caverns. To investigate crack propagation patterns of the surrounding rock of gas storage caverns under high internal pressure, an inelastic fracture phase-field method is developed based on the classical elastic fracture phase-field model. By introducing inelastic strain, the inelastic strain energy density function is adopted as the driving force for phase-field damage evolution, thereby governing crack damage development. The proposed method is applied to simulate crack propagation in an asymmetric notched specimen under tensile stress, and the simulation results are compared with experimental data to verify the accuracy of the method for calculating crack propagation in elastobrittle materials. As the natural surrounding rock is inherently heterogeneous, a heterogeneity parameter is introduced based on the Weibull distribution function to construct a heterogeneous distribution model for gas storage cavern surrounding rock. The inelastic phase-field method is then employed to calculate the crack distribution in the surrounding rock. The numerical results indicate that surrounding rock heterogeneity markedly influences crack propagation patterns. Under identical internal pressure, surrounding rock with stronger heterogeneity is more prone to crack initiation. Under high internal pressure, highly heterogeneous surrounding rock tends to generate more diffuse cracks. Accordingly, geographical environments with more homogeneous surrounding rock should be prioritized during site selection for underground CAES caverns. During cavern excavation, grouting treatment shall be applied to the surrounding rock to improve its strength and homogeneity and suppress crack propagation.