We present evidence that excesses in Be in polluted white dwarfs (WDs) are the result of accretion of icy exomoons that formed in the radiation belts of giant exoplanets. Here we use excess Be in the white dwarf GALEX J2339-0424 as an example. We constrain the parent body abundances of rock-forming elements in GALEX J2339-0424 and show that the overabundance of beryllium in this WD cannot be accounted for by differences in diffusive fluxes through the WD outer envelope nor by chemical fractionations during typical rock-forming processes. We argue instead that the Be was produced by energetic proton irradiation of ice mixed with rock. We demonstrate that the MeV proton fluence required to form the high Be/O ratio in the accreted parent body is consistent with irradiation of ice in the rings of a giant planet within its radiation belt, followed by accretion of the ices to form a moon that is later accreted by the WD. The icy moons of Saturn serve as useful analogs. Our results provide an estimate of spallogenic nuclide excesses in icy moons formed by rings around giant planets in general, including those in the solar system. While excesses in Be have been detected in two polluted WDs to date, including the WD described here, we predict that excesses in the other spallogenic elements Li and B, although more difficult to detect, should also be observed, and that such detections would also indicate pollution by icy exomoons formed in the ring systems of giant planets.