Large room-temperature magnetoresistance observed for devices composed of self-assembled monolayers of different oligophenylene thiols sandwiched between gold contacts has recently been reported [Z. Xie, S. Shi, F. Liu, D. L. Smith, P. P. Ruden, and C. D. Frisbie, ACS Nano 10, 8571 (2016)10.1021/acsnano.6b03853]. The transport mechanism through the organic molecules was determined to be nonresonant tunneling. To explain this kind of magnetoresistance, we develop an analytical model based on the interaction of the tunneling charge carrier with an unpaired charge carrier populating a contact-molecule interface state. The Coulomb interaction between carriers causes the transmission coefficients to depend on their relative spin orientation. Singlet and triplet pairing of the tunneling and the interface carriers thus correspond to separate conduction channels with different transmission probabilities. Spin relaxation enabling transitions between the different channels, and therefore tending to maximize the tunneling current for a given applied bias, can be suppressed by relatively small magnetic fields, leading to large magnetoresistance. Our model elucidates how the Coulomb interaction gives rise to transmission probabilities that depend on spin and how an applied magnetic field can inhibit transitions between different spin configurations.