A three-dimensional Monte Carlo model has been developed to study the transverse beam spreading effect of incident energetic auroral protons during their precipitation in the Earth's upper atmosphere. Energetic protons with an isotropic angular distribution are injected at 700 km altitude. Two types of incident energy spectra, a monoenergetic and a Maxwellian distribution, are considered. Interaction of fast particles with a three-species atmosphere (O, N2, and O2) is included through charge exchange, electron stripping, ionization, excitation, and elastic scattering collisions. A uniform geomagnetic field is assumed in the model. The spreading effect is simulated for both a fine proton beam and a proton arc of longitudinal and latitudinal extent. It is found that the main dispersion region for a fine proton beam is located in the altitude range of around 250--450 km, where the first few charge exchange collisions play a significant role. In the spreading study for a proton arc, we compare the numerical results with previous studies and give a convincing explanation by analyzing atmospheric scale heights and cross-section data. For the purpose of the model validity check, we make a comparison of the Monte Carlo simulation with observations and the results from other models.