Internal processes operating within low-density satellites, although they are poorly understood at present, are extremely significant in determining the scale and type of surface processes and features and the geologic evolution of such satellites. The two Galilean satellites Ganymede and Callisto are examples of such low-density (&rgr;<2) bodies, and they are likely to have undergone a thermal evolution which resulted in varying interior states and surface processes as a function of time. A variety of mechanisms can produce melting of the interior in the early history of such bodies and lead to a configuration characterized by a predominantly water ice lithosphere overlying a mantle containing liquid water. The lithospheric processes operating during such a configuration and the long-term stability of such a configuration are examined. We use the thermal models of Consolmagno and Lewis (1976, 1977, 1978) (which envision significant melting and subsequent thermal evolution of the interior) for the Galilean satellites Ganymede and Callisto to assess (1) sources of stress in the lithosphere, (2) sources of instability of the lithosphere, (3) volcanic processes, and (4) residence time of surface topography. According to these thermal models the lithosphere of Ganymede-like bodies will be relatively thin (<100 km) at the time of melting and be dominated by Ice-I. For bodies like Callisto the lithosphere may be thicker (~250 km) and contain an appreciable thickness of the denser phases of ice. If the lithosphere contains a sufficiently high mass fraction of silicates, gravitational instability taking the form of vertical subsidence of lithospheric blocks into the underlying liquid water could result in renewal of the lithosphere. This instability due to the weight of silicates within the lithosphere should be a one-time occurrence early in the history of such bodies. Renewal due to phase changes in a thick lithosphere could continue as long as the mantle was liquid on bodies like Callisto. Lithospheric instability may have been manifested as vertical movement and 'oceanization' as well as lateral movement and lithospheric overturn analogous to terrestrial plate tectonics. Primary sources of stress in the lithosphere include (1) tidal deformation, which presently produces membrane stresses only on the order of 1--2 bars but which could produce significant stresses in early periods of nonsynchronous rotation and high orbital eccentricity, and (2) volume changes due to phase changes and thermal expansion, which could cause significant stresses (on the order of a kilobar) if the lithosphere deforms elastically. An abundant supply of magma (liquid water) should evolve very early in the history, and surface volcanism should be very important during the initial heating of the interior and expansion of the bodies. Volcanism may be restricted at present owing to the inability of magma to penetrate the upper part of the lithosphere. Dynamical considerations (impact cratering, lithospheric overturn) can cause magma to reach the surface locally in abundance at certain times throughout the history. The long-term stability of a predominantly water ice lithosphere is uncertain. Reynolds and Cassen (1979) have proposed that solid state convection will lead to partial to total mantle refreezing. We present an additional mechanism, related to ice diapirism, which also appears capable of transferring interior heat and freezing the mantle. Observations from the Voyager mission will help to unravel questions concerning the role of early melting, the duration and surface effects of such melting, and the mechanisms, evolution and surface effects of subsequent mantle refreezing and solid state convection.