Inflated membrane structures, useful in vibration/shock isolation devices, terrestrial and space structures, etc., rely on the internal dissipation in the membrane for vibration attenuation. In this work, using the Christensen viscoelastic material model, we study the contact mechanics, displacement-controlled relaxation response, force-controlled creep response, dynamic contact, and energy dissipation due to oscillatory contact in an inflated spherical nonlinear viscoelastic membrane. We consider an inflated spherical membrane squeezed between two large rigid, frictionless, parallel plates. The effective stiffness and damping in the membrane–plate assembly are determined, and a phenomenological model is developed. Under oscillatory contact condition, the energy dissipation per cycle is determined. Further, using the free-vibration test, the damped natural frequency of the membrane–plate system is calculated.