With the equations of state provided by the newly developed density-dependent relativistic Hartree-Fock (DDRHF) theory for hadronic matter, the properties of the static and -equilibrium neutron stars without hyperons are studied for the first time and compared to the predictions of the relativistic mean-field models and recent observational data. The influences of Fock terms on properties of asymmetric nuclear matter at high densities are discussed in detail. Because of the significant contributions from the σ- and ω-exchange terms to the symmetry energy, large proton fractions in neutron stars are predicted by the DDRHF calculations, which strongly affect the cooling process of the star. A critical mass of about , close to the limit of determined by modern soft X-ray data analysis, is obtained by DDRHF with the effective interactions PKO2 and PKO3 for the occurrence of the direct Urca process in neutron stars. The maximum masses of neutron stars given by the DDRHF calculations lie between and , which are in reasonable agreement with the high pulsar mass of from PSR B1516 02B. It is also found that the mass-radius relations of neutron stars determined by DDRHF are consistent with the observational data from thermal radiation measurements in the isolated neutron star RX J1856, quasiperiodic brightness oscillations in the low-mass X-ray binaries 4U 0614 09 and 4U 1636-536, and the redshift determined in the low-mass X-ray binary EXO 0748-676.