We present searches for ultra-high energy (UHE) neutrinos ($> 0.1\,\text{EeV}$) with the Pierre Auger Observatory, following up binary black hole (BBH) mergers detected by the LIGO and Virgo detectors via gravitational waves (GWs). In this work, the so-far published BBH~mergers are combined as standard candles with a hypothetical isotropic UHE~neutrino luminosity $L(t-t_0)$ as a function of the time after the respective merger, $t-t_0$. The UHE~neutrino emission spectrum is assumed to follow a power law distribution $\propto{}E_\nu^{-2}$. Using these assumptions, $L(t-t_0)$ is probed, taking into account the instantaneous effective area of the Pierre Auger Observatory to UHE~neutrinos and the 3D sky localizations of the sources. No UHE~neutrino candidates have been found and upper limits on $L(t-t_0)$ are obtained for the hypothetical cases of emissions lasting 24 hours and 60 days after the merger, respectively. The corresponding upper limit on the total energy per source emitted in UHE~neutrinos does not depend on the emission duration and demonstrates the competitiveness of the Pierre Auger Observatory with dedicated neutrino telescopes.