We present an analysis of MicroBooNE data with a signature of one muon, no pions, and at least one proton above a momentum threshold of 300 MeV/c (CC0$\pi$Np). This is the first differential cross section measurement of this topology in neutrino-argon interactions. We achieve a significantly lower proton momentum threshold than previous carbon and scintillator-based experiments. Using data collected from a total of approximately $1.6 \times 10^{20}$ protons-on-target, we measure the muon neutrino cross section for the CC0$\pi$Np interaction channel in argon at MicroBooNE in the Booster Neutrino Beam which has a mean energy of around 800 MeV. We present the results from a data sample with estimated efficiency of 29\% and purity of 76\% as differential cross sections in five reconstructed variables: the muon momentum and polar angle, the leading proton momentum and polar angle, and the muon-proton opening angle. We include smearing matrices that can be used to "forward-fold" theoretical predictions for comparison with these data. We compare the measured differential cross sections to a number of recent theory predictions demonstrating largely good agreement with this first-ever data set on argon.