  We consider multi-messenger constraints on very heavy dark matter (VHDM) from
recent Fermi gamma-ray and IceCube neutrino observations of isotropic
background radiation. Fermi data on the diffuse gamma-ray background (DGB)
shows a possible unexplained feature at very high energies (VHE), which we have
called the "VHE Excess" relative to expectations for an attenuated power law
extrapolated from lower energies. We show that VHDM could explain this excess,
and that neutrino observations will be an important tool for testing this
scenario. More conservatively, we derive new constraints on the properties of
VHDM for masses of 10^3-10^10 GeV. These generic bounds follow from cosmic
energy budget constraints for gamma rays and neutrinos that we developed
elsewhere, based on detailed calculations of cosmic electromagnetic cascades
and also neutrino detection rates. We show that combining both gamma-ray and
neutrino data is essential for making the constraints on VHDM properties both
strong and robust. In the lower mass range, our constraints on VHDM
annihilation and decay are comparable to other results; however, our
constraints continue to much higher masses, where they become relatively
stronger.
