  Starting from the relativistic realistic nucleon-nucleon ($NN$) interactions,
a newly developed relativistic \textit{ab initio} method, i.e., the
relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space is
employed to study the neutron star properties. First, the one-to-one
correspondence relation for gravitational redshift and mass is established and
used to infer the mass of isolated neutron stars combining the gravitational
redshift measurements. Next, the ratio of the moment of inertia $I$ to $MR^2$
as a function of the compactness $M/R$ is obtained, which is consistent with
the universal relations in the literature. The moment of inertia for
$1.338M_\odot$ pulsar PSR J0737-3039A $I_{1.338M_\odot}$ is predicted to be
1.356$\times10^{45}$, 1.381$\times10^{45}$, and $1.407\times10^{45}\
\mathrm{g~cm^2}$ by the RBHF theory in the full Dirac space with $NN$
interactions Bonn A, B, and C, respectively. Finally, the quadrupole moment of
neutron star is calculated under the slow-rotation and small-tidal-deformation
approximation. The equation of states constructed by the RBHF theory in the
full Dirac space, together with those by the projection method and
momentum-independence approximation, conform to universal $I$-Love-$Q$
relations as well. By combing the tidal deformability from GW170817 and the
universal relations from relativistic \textit{ab initio} methods, the moment of
inertia of neutron star with 1.4 solar mass is also deduced as
$I_{1.4M_\odot}=1.22^{+0.40}_{-0.25}\times 10^{45}\mathrm{g\ cm^2}$.
