We have searched for extremely high energy neutrinos using data taken with the IceCube detector between May 2010 and May 2012. Two neutrino-induced particle shower events with energies around 1 PeV were observed, as reported previously. In this work, we investigate whether these events could originate from cosmogenic neutrinos produced in the interactions of ultrahigh energy cosmic rays with ambient photons while propagating through intergalactic space. Exploiting IceCube's large exposure for extremely high energy neutrinos and the lack of observed events above 100 PeV, we can rule out the corresponding models at more than 90% confidence level. The model-independent quasidifferential 90% C. L. upper limit, which amounts to E-2 phi(nu e)+(nu mu)+(nu tau) = 1.2 x 10(-7) GeV cm(-2) s(-1) sr(-1) at 1 EeV, provides the most stringent constraint in the energy range from 10 PeV to 10 EeV. Our observation disfavors strong cosmological evolution of the highest energy cosmic-ray sources such as the Fanaroff-Riley type II class of radio galaxies.