With the characteristics of low fracturing pressure, little damage to the reservoirs, and assuming the role of carbon storage, supercritical carbon dioxide (SC-CO2) fracturing is suitable for the development of unconventional oil and gas resources. Based on the tensile failure mechanism of rocks, this paper establishes a thermal-fluid-solid coupling initiation pressure model for SC-CO2 fracturing. Using this model, the changes in formation temperature and pore pressure near a wellbore caused by invasion of CO2 into the formation are analyzed, as well as the impact of these changes on the tangential stress of reservoir rocks. The field data of SC-CO2 fracturing in a sandstone gas well are used to validate the reliability of the model. The results show that SC-CO2 fracturing can significantly reduce the initiation pressure, which decreases with the increase in fracturing fluid injection rate. The minimum value of tangential stress is located at the well wall, and the direction of tangential stress caused by formation temperature and pore pressure is opposite, with the former greater than the latter. The increase in Poisson’s ratio, the increase in elastic modulus and the decrease in bottom hole temperature can reduce the initial fracturing pressure of the reservoir. The computation model established in this paper provides an effective method for understanding the reservoir fracturing mechanism under the condition of SC-CO2 invasion.