Increased fragmentation caused by habitat loss presents a major threat to the persistence of animal populations. Whereas the negative effects of habitat loss on biodiversity are well-known, the effects of fragmentation per se on population dynamics and ecosystem stability remain less understood. How fragmentation affects populations is strongly determined by the rate at which individuals can move between separated habitat patches within the fragmented landscape. Here, we use a computational, spatially explicit predator-prey model to investigate how the interplay between fragmentation per se and optimal foraging behavior influences predator-prey interactions and, ultimately, ecosystem stability. We study cases where prey occupies isolated habitat patches and let predators disperse between patches following a Lévy random walk. Our results show that both the Lévy exponent and the degree of fragmentation strongly determine coexistence probabilities. Brownian and ballistic predators go extinct in highly fragmented landscapes and only scale-free predators can coexist with prey. Furthermore, our results reveal that predation causes irreversible loss of prey habitat in highly fragmented landscapes due to the overexploitation of smaller patches. Moreover, our results show that predator movement can reduce, but not prevent not minimize, the amount of irreversibly lost habitat. Our results suggest that incorporating optimal foraging theory into population- and landscape ecology models is crucial to assess the impact of fragmentation on biodiversity and ecosystem stability.