We consider the problem of characterizing and assessing the voltage stability in power distribution networks. Different from previous formulations, we consider the branch-flow parametrization of the power system state, which is particularly effective for radial networks. Our approach to the voltage stability problem is based on a local, approximate, yet highly accurate characterization of the determinant of the power flow Jacobian. Our determinant approximation allows us to construct a voltage stability index that can be computed in a fully scalable and distributed fashion. We provide an upper bound on the approximation error, and we show how the proposed index outperforms other voltage indices that have been recently proposed in the literature.