This paper presents a Robust Adaptive Backstepping Impedance Control (RABIC) strategy for robots operating in contact-rich and uncertain environments. The proposed control strategy considers the complete coupled dynamics of the system and explicitly accounts for key sources of uncertainty, including external disturbances and unmodeled dynamics, while not requiring the robot's dynamic parameters in implementation. We propose a backstepping-based adaptive impedance control scheme for the inner loop to track the reference impedance model. To handle uncertainties, we employ a Taylor series-based estimator for system dynamics and an adaptive estimator for determining the upper bound of external forces. Stability analysis demonstrates the semi-global practical finite-time stability of the overall system. To demonstrate the effectiveness of the proposed method, a simulated mobile manipulator scenario and experimental evaluations on a real Franka Emika Panda robot were conducted. The proposed approach exhibits safer performance compared to PD control while ensuring trajectory tracking and force monitoring. Overall, the RABIC framework provides a solid basis for future research on adaptive and learning-based impedance control for coupled mobile and fixed serially linked manipulators.