We report on the first measurement of flux-integrated single differential cross sections for charged-current (CC) muon neutrino (ν_{μ}) scattering on argon with a muon and a proton in the final state, ^{40}Ar (ν_{μ},μp)X. The measurement was carried out using the Booster Neutrino Beam at Fermi National Accelerator Laboratory and the MicroBooNE liquid argon time projection chamber detector with an exposure of 4.59×10^{19} protons on target. Events are selected to enhance the contribution of CC quasielastic (CCQE) interactions. The data are reported in terms of a total cross section as well as single differential cross sections in final state muon and proton kinematics. We measure the integrated per-nucleus CCQE-like cross section (i.e., for interactions leading to a muon, one proton, and no pions above detection threshold) of (4.93±0.76_{stat}±1.29_{sys})×10^{-38} cm^{2}, in good agreement with theoretical calculations. The single differential cross sections are also in overall good agreement with theoretical predictions, except at very forward muon scattering angles that correspond to low-momentum-transfer events.