A distributed robust adaptive control framework is proposed for an adaptive cruise control system. The proposed approach is designed based on the model reference adaptive control approach. A robust control term is employed to make the system robust to any bounded disturbances, and a concurrent learning framework is leveraged to ensure the convergence of estimated parameters. The main feature of the developed robust adaptive cruise controller is that it does not require the speed of the lead vehicle. It also considers uncertainties in both position and speed in the double integrator model. The string stability notion of the proposed approach is also investigated, and the performance of the control framework is evaluated in simulations in the presence of parametric uncertainties, disturbances, and noise.