Direct current magnetron sputter epitaxy with a liquid Ga sputtering target has been used to grow single-crystal GaN(0001̄) nanorods directly on Si(111) substrates at different working pressures ranging from 5 to 20 mTorr of pure N2. The as-grown GaN nanorods exhibit very good crystal quality from bottom to top without stacking faults, as determined by transmission electron microscopy. The crystal quality is found to increase with increasing working pressure. X-ray diffraction results show that the rods are highly (0001)-oriented. The nanorods exhibit an N-polarity, as determined by convergent beam electron diffraction methods. Sharp and well-resolved 4 K photoluminescence peaks at ~3.474 eV with a FWHM ranging from 1.7 meV to 35 meV are attributed to the intrinsic GaN band edge emission and corroborate the superior structural properties of the material. Texture measurements reveal that the rods have random in-plane orientation when grown on Si(111) with native oxide, while they have an in-plane epitaxial relationship of GaN[112̅0]//Si[11̄0] when grown on substrates without surface oxide.