The supercurrent field effect is experimentally realized in various nanoscale devices, based on the superconductivity suppression by external electric fields being effective for confined systems. In spite of intense research, a microscopic theory of this effect is missing. Here, a microscopic theory of phonon-mediated superconductivity in thin films under an external electric field is presented, which accounts for the effect of quantum confinement on the electronic density of states, on the Fermi energy, and on the electron Coulomb repulsion. By accounting for the complex interplay between quantum confinement, the external static electric field, the Thomas-Fermi screening in the electron-phonon matrix element, and the effect of confinement on the Coulomb repulsion parameter, the theory predicts the critical value of the external electric field as a function of the film thickness, above which superconductivity is suppressed. In particular, this critical value of the electric field is exponentially lower the thinner the film, in agreement with recent experimental observations. Crucially, this effect is predicted by the theory when both Thomas-Fermi screening and the Coulomb pseudopotential are taken into account, along with the respective dependence on the thin film thickness. This microscopic theory opens up new possibilities for the supercurrent field effect and for electric-field gated quantum materials.