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Path following is a fundamental problem in skid-steered mobile robots (SSMR). In this study, a Lyapunov stable curved path following controller was designed to generate the steering control command for an SSMR. In contrast to the existing path following controller design methods, where the complete dynamic model of the robot is considered or not, the steering dynamic characteristics approximated by a first-order model are considered in this study. Together with the kinematic model, a steering control law for following a curved path is designed by using the backstepping technique and Lyapunov stability theory. The proposed method was verified on a real SSMR platform to realize following the straight-line, square, and circular paths. Compared with the steering control law that does not consider the steering dynamics, the proposed method can make the robot converge to the predefined path faster with a smaller error overshoot.