In this letter, a reconfigurable intelligent surface (RIS)-enabled successive interference cancellation (SIC)-based spectrum sharing (SS) scheme is proposed for SS networks where multiple transceiver pairs communicate over the same spectrum at the same time. The key idea is to apply the SIC technique at each receiver and utilize RIS simultaneously, so as to reduce the interference among different transceiver pairs and to focus energy. Specifically, to minimize the total power consumption at all the transmitters under given data rate constraints, the decoding order at each receiver, the reflection coefficients at RIS, and the transmit power at each transmitter are jointly optimized. The formulated problem is a mixed-combinatorial-and-continuous optimization problem and thus is difficult to solve. To address this, a low-complexity but effective decoding order determining method is first proposed. Given the decoding orders, the reflection coefficients and the transmit power are alternately optimized with the help of the semidefinite relaxation method. Substantial performance gains are shown compared with traditional RIS-enabled SS scheme without SIC.