Source localization is one of the essential tasks for underwater acoustic sensor networks (UASNs). The sound speed in water environments varies with depth, therefore the sound waves do not propagate along a straight line, this phenomenon is described as stratification effect. Some range-based UASNs localization algorithms with stratification compensation have been proposed. All these algorithms need to measure the propagation delays for exchanged messages among the nodes. However, these time-based algorithms usually suffer from synchronization errors caused by the node clock skew and clock offset. Thus, we propose a received signal strength (RSS)-based UASNs localization algorithm with stratification compensation (NRA-WSE). In NRA-WSE, urick propagation model is introduced to characterize the RSS, and the stratification effect is modeled by using ray tracing method, then, we adopt the Newton-Raphson algorithm to solve the source location. Finally, we compare our solution to the localization method based on a straight-line propagation model (NRA-NSE). Simulation results show that NRA-WSE performs superbly and meets the Cramér-Rao lower bound (CRLB).