In the present paper, the proton velocity distribution function (VDF) in the solar wind is determined by numerically solving the kinetic evolution equation. We compare the results obtained when considering the effects of external forces and Coulomb collisions with those obtained by adding effects of Alfvén wave turbulence. We use Fokker-Planck diffusion terms to calculate the Alfvénic turbulence, which take into account observed turbulence spectra and kinetic effects of the finite proton gyroradius. Assuming a displaced Maxwellian for the proton VDF at the simulation boundary at 14 solar radii, we show that the turbulence leads to a fast (within several solar radii) development of the anti-sunward tail in the proton VDF. Our results provide a natural explanation for the nonthermal tails in the proton VDFs, which are often observed in-situ in the solar wind beyond 0.3 AU.