Rate-Splitting Multiple Access (RSMA), relying on multi-antenna Rate-Splitting (RS) techniques, has emerged as a powerful strategy for multi-user multi-antenna systems. In this paper, RSMA is introduced as a unified multiple access for multi-antenna radar-communication (RadCom) system, where the base station has a dual communication and radar capability to simultaneously communicate with downlink users and probe detection signals to azimuth angles of interests. Using RS, messages are split into common and private parts, then encoded into common and private streams before being precoded and transmitted. We design the message split and the precoders for this RadCom system such that the Weighted Sum Rate (WSR) is maximized and the transmit beampattern is approximated to the desired radar beampattern under an average transmit power constraint at each antenna. We then propose a framework based on Alternating Direction Method of Multipliers (ADMM) to solve the complicated non-convex optimization problem. Results highlight the benefits of RSMA to unify RadCom transmissions and to manage the interference among radar and communications, over the conventional Space-Division Multiple Access (SDMA) technique.