We demonstrate that the exciton and biexciton emission energies as well as exciton fine structure splitting (FSS) in single (In,Ga)As/GaAs quantum dots (QDs) can be efficiently tuned using hydrostatic pressure in situ in an optical cryostat at up to 4.4 GPa. The maximum exciton emission energy shift was up to 380 meV, and the FSS was up to 180 $\mu$eV. We successfully produced a biexciton antibinding-binding transition in QDs, which is the key experimental condition that generates color- and polarization-indistinguishable photon pairs from the cascade of biexciton emissions and that generates entangled photons via a time-reordering scheme. We perform atomistic pseudopotential calculations on realistic (In,Ga)As/GaAs QDs to understand the physical mechanism underlying the hydrostatic pressure-induced effects.