  We quantify the impact of unpolarized lepton-proton and lepton-nucleus
inclusive deep-inelastic scattering (DIS) cross section measurements from the
future Electron-Ion Collider (EIC) on the proton and nuclear parton
distribution functions (PDFs). To this purpose we include neutral- and
charged-current DIS pseudodata in a self-consistent set of proton and nuclear
global PDF determinations based on the NNPDF methodology. We demonstrate that
the EIC measurements will reduce the uncertainty of the light quark PDFs of the
proton at large values of the momentum fraction $x$, and, more significantly,
of the quark and gluon PDFs of heavy nuclei, especially at small and large $x$.
We illustrate the implications of the improved precision of nuclear PDFs for
the interaction of ultra-high energy cosmic neutrinos with matter.
