  We constrain the neutrino-dark matter cross section using properties of the
dark matter density profiles of Milky Way dwarf spheroidal galaxies. The
constraint arises from core-collapse supernova neutrinos scattering on dark
matter as a form of energy injection, allowing the transformation of the dark
matter density profile from a cusped profile to a flatter profile. We assume a
standard cosmology of dark energy and cold, collisionless, and
non-self-interacting dark matter. By requiring that the dark matter cores do
not lose too much mass or overshoot constraints from stellar kinematics, we
place an upper limit on the cross section of $\sigma_{\nu-\mathrm{DM}}(E_\nu=15
\, \mathrm{MeV}, m_\chi\lesssim130 \, \mathrm{GeV}) \approx 3.4 \times 10^{-23}
\, \mathrm{cm^2}$ and $\sigma_{\nu-\mathrm{DM}}(E_\nu=15 \, \mathrm{MeV},
m_\chi\gtrsim130 \, \mathrm{GeV}) \approx 3.2 \times 10^{-27} \left(
\frac{m_\chi}{1\,\mathrm{GeV}}\right)^2\, \mathrm{cm^2}$, which is stronger
than previous bounds for these energies. Consideration of baryonic feedback or
host galaxy effects on the dark matter profile can strengthen this constraint.
