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 $Z_{mutau}$ 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 $Z_{mutau}$ 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.
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