Charging stations operating in the vehicle-to-grid (V2G) mode can provide effective support for power system operation. However, under grid fault conditions, unbalanced power on the DC side may cause the DC voltage of the charging station to exceed its limits, leading to the disconnection of the charging station from the grid. However, direct disconnection prevents the charging station from supporting the grid during faults and may result in frequent short-term start-stop operations, thereby endangering the power electronic devices within the charging station. To address this issue, the power boundaries of charging stations that avoid DC voltage and AC current limit violations are characterized, and a fault ride-through power feasible region considering fault duration is established. The analysis reveals that the fault ride-through power feasible region shrinks as the fault duration increases, and a method for calculating the critical fault duration that avoids DC voltage limit violations is proposed. Furthermore, the maximum voltage support point of the charging station that avoids AC current and DC voltage limit violations is analytically derived. By considering the relationship between the critical fault duration and grid protection action times, an adaptive fault ride-through control method of charging stations with voltage support capability is proposed. Simulation results show that the proposed method can adopt to different fault conditions, ensuring charging station safety while maximizing voltage support at the point of common coupling.