  Cosmic ray protons generate gamma-rays, neutrinos, and secondary electrons
and positrons (e+/-) through pion-producing collisions with gas atoms. Any
synchrotron or Inverse Compton (IC) radiation from secondary e+/- is therefore
accompanied by pionic gamma-rays. Using the extragalactic gamma-ray background,
we constrain the contribution of secondary e+/- to the cosmic radio, X-ray, and
soft gamma-ray backgrounds. These bounds depend on the unknown hadronic
contribution to the gamma-ray background and the backgrounds' source redshifts.
With our assumptions, we find that IC-upscattered light from secondaries is <~
1/4 of the MeV - GeV gamma-ray background and <~ 10% of the 0.5 keV - 1 MeV
background (for sources at a redshift z <~ 10). The low intensity of the
observed gamma-ray background is marginally inconsistent with a secondary e+/-
origin for the radio background reported by ARCADE at ~3 GHz, unless the
magnetic field strength in their sources is milliGauss or greater. These limits
on the magnetic field strength are sensitive to uncertainties. However, any
contribution to the gamma-ray background from sources not responsible for the
ARCADE excess increases the inconsistency.
