We employ a coupled model for ice-sheet dynamics and Maxwell viscoelastic solid-Earth dynamics, including a gravitationally consistent description of sea level. With this model, we study the influence of the solid Earth on the future evolution of the West Antarctic Ice Sheet (WAIS). Starting from steady-state conditions close to the present-day configuration of the Antarctic Ice Sheet, we apply different atmospheric and oceanic forcings and solid-Earth rheologies in order to analyse the retreat of the WAIS. Climate forcing is the primary control on the occurrence of WAIS collapse. For moderate climate forcing and weak solid-Earth rheologies, however, we find that the relative sea level (RSL) fall associated with the viscoelastic solid-Earth response due to unloading by WAIS retreat limits the retreat to the Amundsen Sea embayment on time scales of several millennia, whereas stiffer Earth structures yield a collapse under these conditions. Under stronger climate forcing, weak Earth structures associated with the West Antarctic rift system produce a delay of up to 5000 years in comparison to stiffer, Antarctic-average solid-Earth rheologies. Furthermore, we find that sea-level rise from an assumed fast deglaciation of the Greenland Ice Sheet induces WAIS collapse in the presence of higher asthenosphere viscosities in cases when the climatic forcing is too weak to force WAIS collapse alone.