We consider a rate-splitting multiple access (RSMA)-based communication and radar coexistence (CRC) system. The proposed system allows an RSMA-based communication system to share spectrum with multiple radars. Furthermore, RSMA enables flexible and powerful interference management by splitting messages into common parts and private parts to partially decode interference and partially treat interference as noise. The RSMA-based CRC system thus significantly improves spectral efficiency, energy efficiency and quality of service (QoS) of communication users (CUs). However, the RSMA-based CRC system raises new challenges. Due to the spectrum sharing, the communication network and the radars cause interference to each other, which reduces the signal-to-interference-plus-noise ratio (SINR) of the radars as well as the data rate of the CUs in the communication network. Therefore, a major problem is to maximize the sum rate of the CUs while guaranteeing their QoS requirements of data transmissions and the SINR requirements of multiple radars. To achieve these objectives, we formulate a problem that optimizes i) the common rate allocation to the CUs, transmit power of common messages and transmit power of private messages of the CUs, and ii) transmit power of the radars. The problem is non-convex with multiple decision parameters, which is challenging to be solved. We propose two algorithms. The first sequential quadratic programming (SQP) can quickly return a local optimal solution, and has been known to be the state-of-the-art in nonlinear programming methods. The second is an additive approximation scheme (AAS) which solves the problem globally in a reasonable amount of time, based on the technique of applying exhaustive enumeration to a modified instance. The simulation results show the improvement of the AAS compared with the SQP in terms of sum rate.