  Recent progress of nucleosynthesis work as well as the discovery of a
kilonova associated with the gravitational-wave source GW170817 indicates that
neutron star mergers (NSM) can be a site of the r-process. Several studies of
galactic chemical evolution, however, have pointed out inconsistencies between
this idea and the observed stellar abundance signatures in the Milky Way: (a)
the presence of Eu at low (halo) metallicity and (b) the descending trend of
Eu/Fe at high (disc) metallicity. In this study, we explore the galactic
chemical evolution of the Milky Way's halo, disc and satellite dwarf galaxies.
Particular attention is payed to the forms of delay-time distributions for both
type Ia supernovae (SN Ia) and NSMs. The Galactic halo is modeled as an
ensemble of independently evolving building-block galaxies with different
masses. The single building blocks as well as the disc and satellite dwarfs are
treated as well-mixed one-zone systems. Our results indicate that the
aforementioned inconsistencies can be resolved and thus NSMs can be the unique
r-process site in the Milky Way, provided that the delay-time distributions
satisfy the following conditions: (i) a long delay (~1 Gyr) for the appearance
of the first SN Ia (or a slow early increase of its number) and (ii) an
additional early component providing >~ 50% of all NSMs with a delay of ~0.1
Gyr. In our model, r-process-enhanced and r-process-deficient stars in the halo
appear to have originated from ultra-faint dwarf-sized and massive building
blocks, respectively. Our results also imply that the natal kicks of binary
neutron stars have a little impact on the evolution of Eu in the disc.
