The simplest unified extension of the minimal supersymmetric standard model with bilinear R-parity violation naturally predicts a hierarchical neutrino mass spectrum, in which one neutrino acquires mass by mixing with neutralinos, while the other two get mass radiatively. We have performed a full one-loop calculation of the neutralino-neutrino mass matrix in the bilinear R-p minimal supersymmetric standard model, taking special care to achieve a manifestly gauge invariant calculation. Moreover we have performed the renormalization of the heaviest neutrino, needed in order to get meaningful results. The atmospheric mass scale and maximal mixing angle arise from tree-level physics, while solar neutrino scale and oscillations follow from calculable one-loop corrections. If universal supergravity assumptions are made on the soft-supersymmetry breaking terms then the atmospheric scale is calculable as a function of a single R-p violating parameter by the renormalization group evolution due to the nonzero bottom quark Yukawa coupling. The solar neutrino problem must be accounted for by the small mixing angle Mikheyev-Smirnov-Wolfenstein (MSW) solution. If these assumptions are relaxed then one can implement large mixing angle solutions. The theory predicts the lightest supersymmetic particle decay to be observable at high-energy colliders, despite the smallness of neutrino masses indicated by experiment. This provides an independent way to test this solution of the atmospheric and solar neutrino anomalies.