The detection of ultrahigh-energy neutrinos, with energies in the PeV range or above, is a topic of great interest in modern astroparticle physics. The importance comes from the fact that these neutrinos point back to the most energetic particle accelerators in the Universe and provide information about their underlying acceleration mechanisms. Atmospheric neutrinos are a background for these challenging measurements, but their rate is expected to be negligible above ≈1 PeV. In this work we describe the feasibility to study ultrahigh-energy neutrinos based on the Earth-skimming technique, by detecting the charged leptons produced in neutrino-nucleon interactions in a high mass target. We propose to detect the charged leptons, or their decay products, with the High Altitude Water Cherenkov (HAWC) observatory and use as a large-mass target for the neutrino interactions the Pico de Orizaba volcano, the highest mountain in Mexico. In this work we develop an estimate of the detection rate using a geometrical model to calculate the effective area of the observatory. Our results show that it may be feasible to perform measurements of the ultrahigh-energy neutrino flux from cosmic origin during the expected lifetime of the HAWC observatory.