On 2019 August 14, the Advanced LIGO and Virgo interferometers detected the high-significance gravitational wave (GW) signal S190814bv. The GW data indicated that the event resulted from a neutron star-black hole (NSBH) merger, or potentially a low-mass binary BH merger. Due to the low false-alarm rate and the precise localization (23 deg(2) at 90%), S190814bv presented the community with the best opportunity yet to directly observe an optical/near-infrared counterpart to an NSBH merger. To search for potential counterparts, the GROWTH Collaboration performed real-time image subtraction on six nights of public Dark Energy Camera images acquired in the 3 weeks following the merger, covering >98% of the localization probability. Using a worldwide network of follow-up facilities, we systematically undertook spectroscopy and imaging of optical counterpart candidates. Combining these data with a photometric redshift catalog, we ruled out each candidate as the counterpart to S190814bv and placed deep, uniform limits on the optical emission associated with S190814bv. For the nearest consistent GW distance, radiative transfer simulations of NSBH mergers constrain the ejecta mass of S190814bv to be M-ej < 0.04 M-circle dot at polar viewing angles, or M-ej < 0.03 M-circle dot if the opacity is kappa < 2 cm(2)g(-1). Assuming a tidal deformability for the NS at the high end of the range compatible with GW170817 results, our limits would constrain the BH spin component aligned with the orbital momentum to be chi < 0.7 for mass ratios Q < 6, with weaker constraints for more compact NSs.