Electroweak (EW) physics at the future electron-proton colliders LHeC and FCC-eh is studied. Simulated neutral-current and charged-current deep-inelastic scattering cross sections are employed for simultaneous determinations of the parton distribution functions of the proton together with the fundamental parameters of the EW theory, including their statistical and systematic uncertainties. Uncertainties of the $W$ and $Z$ boson masses are determined and compared to uncertainties obtained from HERA combined data. The LHeC data will allow for a determination of $m_{\rm W}$ with an uncertainty of 17 MeV, and the FCC-eh with 10 MeV, thus exceeding the precision of the currently most precise single measurements. The LHeC or FCC-eh data will allow for a precision determination of the vector and axial-vector couplings of the light quarks to the $Z$-boson, with uncertainties being smaller by an order order of magnitude than current measurements. It is shown, that the measurements of the EW parameters are not limited by the precision of the parton distribution functions, which have also to be determined from the same data. The measurements of the inclusive DIS cross sections as a function of the four-momentum transfer squared $Q^2$ will allow for high precision tests of the scale dependence of the EW theory in the range from a few GeV up to the TeV regime from a single process.