We analyze from a far field the evolution of an accelerated interface separating ideal incompressible fluids of different densities. We develop and apply a general matrix method and identify a new fluid instability that occurs only when the acceleration magnitude exceeds a threshold value depending on the fluids' density ratio and uniform velocities and the perturbation wavelength. The dynamics conserves the fluxes of mass, momentum and energy, has potential velocity fields in the bulk, and is shear-free at the interface. The interface stability is set by the interplay of inertia and acceleration. Surface tension may also stabilize the dynamics by a distinct mechanism. The growth rate, the flow fields' structure and stabilization mechanisms of this new fluid instability depart substantially from those of other instabilities, thus suggesting new opportunities for the understanding, diagnostics, and control of interfacial dynamics.