Mechanical face seals are crucial components of automotive cooling water pumps and affect the safe operation of the pump. This article focuses on the effect of friction instabilities on the wear of the seals. Friction instabilities, such as stick-slip, occur when the axle is decelerated or operated at a low speed. Based on previous studies, a simulation model is proposed of a mechanical face seal that considers the interaction of asperities of non-Gaussian surfaces and the heat transfer between the sealing rings. According to the Archard wear equation, a numerical wear simulation is performed, and the wear distance rate and wear time rate are obtained. A comparison of the contact pressure of the Gaussian and non-Gaussian surfaces indicates that the latter is more likely to generate high contact pressure, thereby producing more significant wear. The viscous shear heat and frictional heat due to asperity contact decrease with an increase in the thickness of the tapered film. As the shaft decelerates, the wear distance rate increases with an increase in the axial stiffness. The axial damping only affects the duration of the oscillations. The wear time rate decreases with an increase in the torsional stiffness and torsional damping. The results of this research provide guidelines for estimating the wear of mechanical seals when friction instabilities occur.