  Fermi has resolved several star-forming galaxies, but the vast majority of
the star-forming universe is unresolved and thus contributes to the
extragalactic gamma ray background (EGB). Here, we calculate the contribution
from star-forming galaxies to the EGB in the Fermi range from 100 MeV to 100
GeV, due to inverse-Compton (IC) scattering of the interstellar photon field by
cosmic-ray electrons. We first construct a one-zone model for a single
star-forming galaxy, assuming supernovae power the acceleration of cosmic rays.
The same IC interactions leading to gamma rays also substantially contribute to
the energy loss of the high-energy cosmic-ray electrons. Consequently, a
galaxy's IC emission is determined by the relative importance of IC losses in
the cosmic-ray electron energy budget ("partial calorimetry"). We use our
template for galactic IC luminosity to find the cosmological contribution of
star-forming galaxies to the EGB. For all of our models, we find the IC EGB
contribution is almost an order of magnitude less than the peak of the emission
due to cosmic-ray ion interactions (mostly pionic p_cr p_ism \rightarrow \pi_0
\rightarrow \gamma \gamma); even at the highest Fermi energies, IC is
subdominant. Moreover, the flatter IC spectrum increases the high-energy signal
of the pionic+IC sum, bringing it into better agreement with the EGB spectral
index observed by Fermi . Partial calorimetry ensures that the overall IC
signal is well constrained, with only modest uncertainties in the amplitude and
spectral shape for plausible model choices. Partial calorimetry of cosmic-ray
electrons should hold true in both normal and starburst galaxies, and thus we
include starbursts in our calculation. We conclude with a brief discussion on
how the pionic spectral feature and other methods can be used to measure the
star-forming component of the EGB.
