On September 14, 2015 the Advanced LIGO detectors observed their first gravitational wave (GW) transient GW150914. This was followed by a second GW event observed on December 26, 2015. Both events were inferred to have arisen from the merger of black holes in binary systems. Such a system may emit neutrinos if there are magnetic fields and disk debris remaining from the formation of the two black holes. With the surface detector array of the Pierre Auger Observatory we can search for neutrinos with energy ${E}_{\ensuremath{\nu}}$ above 100 PeV from pointlike sources across the sky with equatorial declination from about $\ensuremath{-}65\ifmmode^\circ\else\textdegree\fi{}$ to $+60\ifmmode^\circ\else\textdegree\fi{}$, and, in particular, from a fraction of the 90% confidence-level inferred positions in the sky of GW150914 and GW151226. A targeted search for highly inclined extensive air showers, produced either by interactions of downward-going neutrinos of all flavors in the atmosphere or by the decays of tau leptons originating from tau-neutrino interactions in the Earth's crust (Earth-skimming neutrinos), yielded no candidates in the Auger data collected within $\ifmmode\pm\else\textpm\fi{}500\text{ }\text{ }\mathrm{s}$ around or 1 day after the coordinated universal time (UTC) of GW150914 and GW151226, as well as in the same search periods relative to the UTC time of the GW candidate event LVT151012. From the nonobservation we constrain the amount of energy radiated in ultrahigh-energy neutrinos from such remarkable events.