The optical responses of 75-150 nm diameter gold nanorings to changes in local refractive index have been quantified by near-infrared extinction spectroscopy and compared to DDA calculations and an analytical approach. The "bulk" refractive index sensitivities of gold nanorings are substantially (>5 times) larger than those of nanodisks with similar diameters. Nanorings retain a significantly larger sensitivity than nanodisks at the same spectral position, demonstrating a clear shape dependence that may correlate to a systematic difference in the influence of the dielectric substrate. The nanoring bulk refractive index sensitivity scales linearly with plasmon peak position. The spectral sensitivity to thin films of alkanethiols gave a shift of approximately 5.2 nm/CH2 unit while bulk sensitivities as high as 880 nm/RIU were observed, the highest such reported sensitivities. Both bulk and thin dielectric film sensitivities correlated well with theory. Real-time label-free monitoring of protein binding via molecular recognition was demonstrated.