We present a search for the diffuse extremely-high-energy neutrino flux using 12.6 years of IceCube data. The nonobservation of neutrinos with energies well above 10 PeV constrains the all-flavor neutrino flux at 1018 eV to a level of E2φνe+νμ+ντ≃10-8 GeV cm-2 s-1 sr-1, the most stringent limit to date. Using these data, we constrain the proton fraction of ultrahigh-energy cosmic rays (UHECRs) above ≃30 EeV to be ≲70% (at 90% CL) if the cosmological evolution of the sources is comparable to or stronger than the star formation rate. This is the first result to disfavor the "proton-only"hypothesis for UHECR in this evolution regime using neutrino data. This result complements direct air-shower measurements by being insensitive to uncertainties associated with hadronic interaction models. We also evaluate the tension between IceCube-s nonobservation and the ∼200 PeV KM3NeT neutrino candidate (KM3-230213A), finding it to be ∼2.9σ based on a joint-livetime fit between neutrino datasets.