The surface tension of a self-rewetting fluid (SRF) has a nonmonotonic variation with the increase of temperature, implying potential applications in many industrial fields. In this paper, flow patterns and stability analysis are numerically performed for a gravity-driven self-rewetting fluid film flowing down a heated vertical plane with wall slip. Using the thin film theory, the evolution equation for the interfacial thickness is derived. The discussion is given considering two cases in the review of the temperature difference between the interfacial temperature and the temperature corresponding to the minimum surface tension. The base state of the two-dimensional flow is first obtained, and the influence of the Marangoni effect and slippery effect is analyzed. Then linear stability analysis and related numerical verification are displayed, showing good consistency with each other. For a low interfacial temperature, the Marangoni promotes the fingering instability, and for a high interfacial temperature, the inverse Marangoni impedes the surface instability. The wall slip is found to influence the free surface in a complex way because it can either destabilize or stabilize the flow of the free surface.