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A recent result from Fermilab suggests that the measured W-boson mass deviates from the prediction of the Standard Model (SM) with a significance of $>7σ$, and there may exist new physics beyond the SM. It is proposed that the inert two Higgs doublet model (i2HDM) can well explain the new W-boson mass. Meanwhile, the lightest neutral scalar $S$ in the i2HDM can be stable and play the role of dark matter with a preferred dark matter mass of $\sim 54-74$ GeV. It is also found that part of the parameter space of this model can explain both the Galactic center GeV gamma-ray excess detected by $Fermi$-LAT and the GeV antiproton excess detected by AMS-02 through a $SS\rightarrow WW^*$ annihilation. In this paper, we aim to test the possible common i2HDM origin of the three anomaly/excesses using the $Fermi$-LAT observations of Milky Way dwarf spheroidal (dSph) galaxies. We perform single and stacking analyses on 19 dSphs that have J-factor measurements. We find that our upper limits are below the favored parameters and seems to be able to exclude the possibility of a common origin of the three anomaly/excesses. However, because the J-factor measurements include relatively large uncertainties, which come from the measurements of stellar kinematics, whether this model could be reliably excluded needs to be further confirmed by future observations.