Nanoscale electron transport through the purple membrane monolayer, a two-dimensional crystal lattice of the transmembrane protein bacteriorhodopsin, is studied by conductive atomic force microscopy. We demonstrate that the purple membrane exhibits nonresonant tunneling transport, with two characteristic tunneling regimes depending on the applied voltage (direct and Fowler-Nordheim). Our results show that the purple membrane can carry significant current density at the nanometer scale, several orders of magnitude larger than previously estimated by macroscale measurements.