We study a velocity-space quasilinear diffusion of the solar wind protons driven by oblique Alfven turbulence at proton kinetic scales. Turbulent fluctuations at these scales possess properties of kinetic Alfven waves (KAWs) that are efficient in Cherenkov resonant interactions. The proton diffusion proceeds via Cherenkov kicks and forms a quasilinear plateau - nonthermal proton tail in the velocity distribution function (VDF). The tails extend in velocity space along the mean magnetic field from 1 to (1.5-3) VA, depending on the spectral break position, turbulence amplitude at the spectral break, and spectral slope after the break. The most favorable conditions for the tail generation occur in the regions where the proton thermal and Alfven velocities are about the same, VTp/VA = 1. The estimated formation times are within 1-2 h for typical tails at 1 AU, which is much shorter than the solar wind expansion time. Our results suggest that the nonthermal proton tails, observed in-situ at all heliocentric distances beyond 0.3 AU, are formed in the solar wind locally by the KAW turbulence. We also suggest that the bump-on-tail features - proton beams, often seen in the proton VDFs, can be formed at a later evolution stage of the nonthermal tails by the time-of-flight effects.