We examine the clustering properties of H I-selected galaxies through an analysis of the H I Parkes All-Sky Survey Catalogue (HICAT) two-point correlation function. Various subsamples are extracted from this catalog to study the overall clustering of H I-rich galaxies and its dependence on luminosity, H I gas mass, and rotational velocity. These samples cover the entire southern sky δ < 0°, containing up to 4174 galaxies over the radial velocity range 300-12,700 km s^-1. A scale length of r₀ = 3.45 ± 0.25 h^-1 Mpc and slope of γ = 1.47 ± 0.08 is obtained for the H I-rich galaxy real-space correlation function, making gas-rich galaxies among the most weakly clustered objects known. H I-selected galaxies also exhibit weaker clustering than optically selected galaxies of comparable luminosities. Good agreement is found between our results and those of synthetic H I-rich galaxy catalogs generated from the Millennium Run cold dark matter (CDM) simulation. Bisecting HICAT using different parameter cuts, clustering is found to depend most strongly on rotational velocity and luminosity, while the dependency on H I mass is marginal. Splitting the sample around v_rot = 108 km s^-1, a scale length of r₀ = 2.86 ± 0.46 h^-1 Mpc is found for galaxies with low rotational velocities compared to r₀ = 3.96 ± 0.33 h^-1 Mpc for the high rotational velocity sample.