High-energy cosmic neutrinos can reveal new fundamental particles and interactions, probing energy and distance scales far exceeding those accessible in the laboratory. This white paper describes the outstanding particle physics questions that high-energy cosmic neutrinos can address in the coming decade. A companion white paper discusses how the observation of cosmic neutrinos can address open questions in astrophysics. Tests of fundamental physics using high-energy cosmic neutrinos will be enabled by detailed measurements of their energy spectrum, arrival directions, flavor composition, and timing.