We study the S01 proton pairing gap in β-equilibrated neutron star matter within the framework of chiral effective field theory. We focus on the role of three-body forces, which strongly modify the effective proton-proton spin-singlet interaction in dense matter. We find that three-body forces generically reduce both the size of the pairing gap and the maximum density at which proton pairing may occur. The pairing gap is computed within Bardeen-Cooper-Schrieffer theory using a single-particle dispersion relation calculated up to second order in perturbation theory. Model uncertainties are estimated by varying the nuclear potential (its order in the chiral expansion and high-momentum cutoff) and the choice of single-particle spectrum in the gap equation. We find that a second-order perturbative treatment of the single-particle spectrum suppresses the proton S01 pairing gap relative to the use of a free spectrum. We estimate the critical temperature for the onset of proton superconductivity to be Tc=(3.2-5.1)×109 K, which is consistent with previous theoretical results in the literature and marginally within the range deduced from a recent Bayesian analysis of neutron star cooling observations.