Adopting the framework of the halo occupation distribution (HOD), we investigate the ability of galaxy clustering measurements to simultaneously constrain cosmological parameters and galaxy bias. Starting with a fiducial cosmological model and galaxy HOD, we calculate spatial clustering observables on a range of length and mass scales, dynamical clustering observables that depend on galaxy peculiar velocities, and the galaxy-matter cross-correlation measurable by weak lensing. We then change one or more cosmological parameters and use χ~2 minimization to find the galaxy HOD that best reproduces the original clustering. Our parameterization of the HOD incorporates a flexible relation between galaxy occupation numbers and halo mass and allows spatial and velocity bias of galaxies within dark matter halos. Despite this flexibility, we find that changes to the HOD cannot mask substantial changes to the matter density Ω_m, the matter clustering amplitude σ_8, or the shape parameter Γ of the linear matter power spectrum: cosmology and bias are not degenerate. With the conservative assumption of 10% fractional errors, the set of observables considered here can provide ～10% (1σ) constraints on σ_8, Ω_m, and Γ, using galaxy clustering data alone. The combination σ_8Ω_m~(0.75) is constrained to ～5%. In combination with traditional methods that focus on large-scale structure in the "perturbative" regime, HOD modeling can greatly amplify the cosmological power of galaxy redshift surveys by taking advantage of high-precision clustering measurements at small and intermediate scales (from sub-Mpc to ～20 hr~(-1) Mpc). At the same time, the inferred constraints on the galaxy HOD provide valuable tests of galaxy formation theory.
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