  (ABRIDGED) We use hydrodynamical simulations of disk galaxies to study
relations between star formation and properties of the molecular interstellar
medium (ISM). We implement a model for the ISM that includes low-temperature
(T<10^4K) cooling, directly ties the star formation rate to the molecular gas
density, and accounts for the destruction of H2 by an interstellar radiation
field from young stars. We demonstrate that the ISM and star formation model
simultaneously produces a spatially-resolved molecular-gas surface density
Schmidt-Kennicutt relation of the form Sigma_SFR \propto Sigma_Hmol^n_mol with
n_mol~1.4 independent of galaxy mass, and a total gas surface density -- star
formation rate relation Sigma_SFR \propto Sigma_gas^n_tot with a power-law
index that steepens from n_tot~2 for large galaxies to n_tot>~4 for small dwarf
galaxies. We show that deviations from the disk-averaged Sigma_SFR \propto
Sigma_gas^1.4 correlation determined by Kennicutt (1998) owe primarily to
spatial trends in the molecular fraction f_H2 and may explain observed
deviations from the global Schmidt-Kennicutt relation.
