Neutrinos offer a window to physics beyond the Standard Model. In particular, high-energy astrophysical neutrinos, with TeV-PeV energies, may provide evidence of new, "secret" neutrino-neutrino interactions that are stronger than ordinary weak interactions. During their propagation over cosmological distances, high-energy neutrinos could interact with the cosmic neutrino background via secret ... [Show full abstract] interactions, developing characteristic energy-dependent features in their observed energy distribution. For the first time, we look for signatures of secret neutrino interactions in the diffuse flux of high-energy astrophysical neutrinos, using 6 years of publicly available IceCube High Energy Starting Events (HESE). We find no significant evidence for secret neutrino interactions, but place competitive upper limits on the coupling strength of the new mediator through which they occur, in the mediator mass range of 1-100 MeV.