In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.