  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
$>7\sigma$, 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.
