Using the IceCube Neutrino Observatory, we search for high-energy neutrino emission coincident with compact binary mergers observed by the LIGO and Virgo gravitational-wave (GW) detectors during their first and second observing runs. We present results from two searches targeting emission coincident with the sky localization of each GW event within a 1000 s time window centered around the reported merger time. One search uses a model-independent unbinned maximum-likelihood analysis, which uses neutrino data from IceCube to search for pointlike neutrino sources consistent with the sky localization of GW events. The other uses the Low-Latency Algorithm for Multi-messenger Astrophysics, which incorporates astrophysical priors through a Bayesian framework and includes LIGO-Virgo detector characteristics to determine the association between the GW source and the neutrinos. No significant neutrino coincidence is seen by either search during the first two observing runs of the LIGO-Virgo detectors. We set upper limits on the time-integrated neutrino emission within the 1000 s window for each of the 11 GW events. These limits range from 0.02 to 0.7 . We also set limits on the total isotropic equivalent energy, E-iso, emitted in high-energy neutrinos by each GW event. These limits range from 1.7 x 10(51) to 1.8 x 10(55) erg. We conclude with an outlook for LIGO-Virgo observing run O3, during which both analyses are running in real time.