Weakly interacting massive particle (WIMP) as a dark matter (DM) candidate is further inspired by recent AMS-02 data, which confirm the excess of positron fraction observed earlier by PAMELA and Fermi-LAT experiments. Additionally, the excess of positron+electron flux is still significant in the measurement of Fermi-LAT. For solving the problem of massive neutrinos and observed excess of cosmic-ray by DM annihilation, we study the model with an inert Higgs doublet (IHD) in the framework of type-II seesaw mechanism by imposing a $Z_2$ symmetry on the IHD, where the lightest particle of IHD is the DM candidate while the neutrino masses origin from the Higgs triplet in type-II seesaw model. We calculate the cosmic-ray production in our model by using three kinds of neutrino mass spectra, classified as normal ordering, inverted ordering and quasi-degeneracy. We find that if leptonic triplet decays are dominant, the observed excess of positron/electron flux could be explained well in normal ordered neutrino mass spectrum, when the constraints of DM relic density and comic-ray antiproton spectrum are taken into account. Moreover, excess of comic-ray neutrinos is implied in our model. We find that our results on $\langle \sigma v \rangle$ are satisfied with and close to the upper limit of IceCube analysis. More data from comic-ray neutrinos could test our model.