This paper aims at understanding the adhesion mechanics of a pressure-controlled adhesive thin elastomeric membrane for soft robotic gripping on non-planar, curved surfaces. The adhesive elastic membrane is lined with gecko-inspired microfiber arrays and can be inflated or deflated by controlled internal air pressure. Previous studies with the soft robotic grippers using dry adhesives showed repeatable adhesion and transfer printing of various non-planar objects with high reliability. In this study, we perform experimental characterization and theoretical analysis to better understand the influence of size and shape of the adhering curved objects on the range of internal air pressures as well as the force profile. In addition, decrease in the internal air pressure results in an increased pull-off force associated with a change in the range of gripper retraction for pulling off the membrane on various curved surfaces. An approximate analytical model dealing with the complex boundary conditions presented in this paper can provide quantitative estimates of pull-off forces for a wide variety of surface curvatures and internal air pressures, as well as qualitative understanding of how force profiles change under moderate pressure differentials.