Current data suggest that the central mass densities ρ0 and phase-space densities Q ≡ ρ0/σ of cosmological halos in the present universe are correlated with their velocity dispersions σV over a very wide range of σV from less than 10 to more than 1000 km s-1. Such correlations are an expected consequence of the statistical correlation of the formation epochs of virialized objects in the cold dark matter (CDM) model with their masses; the smaller mass halos typically form first and merge to form larger mass halos later. We have derived the Q-σV and ρ0-σV correlations for different CDM cosmologies and compared the predicted correlations with the observed properties of a sample of low-redshift halos ranging in size from dwarf spheroidal galaxies to galaxy clusters. Our predictions are generally consistent with the data, with preference for the currently favored, flat low-density CDM model with cosmological constant (ΛCDM). Such a comparison serves to test the basic CDM paradigm while constraining the background cosmology and the power spectrum of primordial density fluctuations, including larger wavenumbers than have previously been constrained.