We explore the parameter space for the contribution to Callisto's H corona observed by the Hubble Space Telescope from sublimated H2O and radiolytically produced H-2 using the Direct Simulation Monte Carlo method. The spatial morphology of this corona produced via photoelectron and magnetospheric electron-impact-induced dissociation is described by tracking the motion of and simulating collisions between the hot H atoms and thermal molecules including a near-surface O-2 component. Our results indicate that sublimated H2O produced from the surface ice, whether assumed to be intimately mixed with or distinctly segregated from the dark nonice or ice-poor regolith, cannot explain the observed structure of the H corona. On the other hand, a global H-2 component can reproduce the observation, and is also capable of producing the enhanced electron densities observed at high altitudes by Galileo's plasma-wave instrument, providing the first evidence of H-2 in Callisto's atmosphere. The range of H-2 surface densities explored, under a variety of conditions, that are consistent with these observations is similar to(0.4-1) x 10(8) cm(-3). The simulated H-2 escape rates and estimated lifetimes suggest that Callisto has a neutral H-2 torus. We also place a rough upper limit on the peak H2O number density (less than or similar to 10(8) cm(-3)), column density (less than or similar to 10(15) cm(-2)), and sublimation flux (less than or similar to 10(12) cm(-2) s(-1)), all of which are 1-2 orders of magnitude less than that assumed in previous models. Finally, we discuss the implications of these results, as well as how they compare to Europa and Ganymede.