Classic local stability theory predicts that complex ecological networks are unstable and are unlikely to persist despite empiricists' abundant documentation of such complexity in nature. This contradiction has puzzled biologists for decades. While some have explored how stability may be achieved in small modules of a few interacting species, rigorous demonstrations of how large complex and ecologically realistic networks dynamically persist remain scarce and inadequately understood. Here, we help fill this void by combining structural models of complex food webs with nonlinear bioenergetic models of population dynamics parameterized by biological rates that are allometrically scaled to populations' average body masses. Increasing predator-prey body mass ratios increase population persistence up to a saturation level that is reached by invertebrate and ectotherm vertebrate predators when being 10 or 100 times larger than their prey respectively. These values are corroborated by empirical predator-prey body mass ratios from a global data base. Moreover, negative effects of diversity (i.e. species richness) on stability (i.e. population persistence) become neutral or positive relationships at these empirical ratios. These results demonstrate that the predator-prey body mass ratios found in nature may be key to enabling persistence of populations in complex food webs and stabilizing the diversity of natural ecosystems.