We show that quantum square ice-namely, the two-dimensional version of proton or spin ice with tunable quantum tunneling of the electric or magnetic dipole moment-exhibits a quantum spin-liquid phase supporting fractionalized spinons. This phase corresponds to a thermally induced, deconfined quantum Coulomb phase of a two-dimensional lattice gauge theory. It emerges at finite, yet exceedingly low temperatures from the melting of two distinct order-by-disorder phases appearing in the ground state: a plaquette valence-bond solid for low tunneling; and a canted Néel state for stronger tunneling. The latter phases appear via the highly nonlinear effect of quantum fluctuations within the degenerate manifold of ice-rule states, and they can be identified as the two competing ground states of a discrete lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory.