We present a measurement of the refractive index of firn ice at Summit Station, the site of the Radio Neutrino Observatory in Greenland (RNO-G). The refractive index profile was determined without relying on density data using absolute time-of-flight measurements with sub-nanosecond precision. We establish the consistency of these findings across several locations near Summit Station and different signal polarizations. From these findings, we derive a parametrized model of the firn ice with uncertainties. Furthermore, we test the consistency of the derived models with additional pulser measurements and compare the results to independent measurements of the density.