  The observation of neutrino masses, mixing and the existence of dark matter
are amongst the most important signatures of physics beyond the Standard Model
(SM). In this paper, we propose to extend the SM by a local $L_\mu - L_\tau$
gauge symmetry, two additional complex scalars and three right-handed
neutrinos. The $L_\mu - L_\tau$ gauge symmetry is broken spontaneously when one
of the scalars acquires a vacuum expectation value. The $L_\mu - L_\tau$ gauge
symmetry is known to be anomaly free and can explain the beyond SM measurement
of the anomalous muon $({\rm g-2})$ through additional contribution arising
from the extra $Z_{\mu\tau}$ mediated diagram. Small neutrino masses are
explained naturally through the Type-I seesaw mechanism, while the mixing
angles are predicted to be in their observed ranges due to the broken
$L_\mu-L_\tau$ symmetry. The second complex scalar is shown to be stable and
becomes the dark matter candidate in our model. We show that while the
$Z_{\mu\tau}$ portal is ineffective for the parameters needed to explain the
anomalous muon $({\rm g-2})$ data, the correct dark matter relic abundance can
easily be obtained from annihilation through the Higgs portal. Annihilation of
the scalar dark matter in our model can also explain the Galactic Centre gamma
ray excess observed by Fermi-LAT. We show the predictions of our model for
future direct detection experiments and neutrino oscillation experiments.
