  We perform a full nuclear-network numerical calculation of the $r$-process
nuclei in binary neutron-star mergers (NSMs), with the aim of estimating
$\gamma$-ray emissions from the remnants of Galactic NSMs up to $10^6$ years
old. The nucleosynthesis calculation of 4,070 nuclei is adopted to provide the
elemental composition ratios of nuclei with an electron fraction $Y_{\rm e}$
between 0.10 and 0.45 . The decay processes of 3,237 unstable nuclei are
simulated to extract the $\gamma$-ray spectra. As a result, the NSMs have
different spectral color in $\gamma$-ray band from various other astronomical
objects at less than $10^5$ years old. In addition, we propose a new
line-diagnostic method for $Y_{\rm e}$ that uses the line ratios of either
$^{137{\rm m}}$Ba/$^{85}$K or $^{243}$Am/$^{60{\rm m}}$Co, which become larger
than unity for young and old $r$-process sites, respectively, with a low
$Y_{\rm e}$ environment. From an estimation of the distance limit for
$\gamma$-ray observations as a function of the age, the high sensitivity in the
sub-MeV band, at approximately $10^{-9}$ photons s$^{-1}$ cm$^{-2}$ or
$10^{-15}$ erg s$^{-1}$ cm$^{-2}$, is required to cover all the NSM remnants in
our Galaxy if we assume that the population of NSMs by
\citet{2019ApJ...880...23W}. A $\gamma$-ray survey with sensitivities of
$10^{-8}$--$10^{-7}$ photons s$^{-1}$ cm$^{-2}$ or $10^{-14}$--$10^{-13}$ erg
s$^{-1}$ cm$^{-2}$ in the 70--4000 keV band is expected to find emissions from
at least one NSM remnant under the assumption of NSM rate of 30 Myr$^{-1}$. The
feasibility of $\gamma$-ray missions to observe Galactic NSMs are also studied.
