  The "blazar simplified view" is a new paradigm that explains well the diverse
statistical properties of blazars observed over the entire electromagnetic
spectrum on the basis of minimal assumptions on blazars' physical and
geometrical properties. In this paper, the fourth in a series, we extend the
predictions of this paradigm below the sensitivity of existing surveys and
estimate the contribution of blazars to the X-ray and gamma-ray extragalactic
backgrounds. We find that the integrated light from blazars can explain up to
100% of the cosmic background at energies larger than ~10 GeV, and contribute
~40 to 70% of the gamma-ray diffuse radiation between 100 MeV and 10 GeV. The
contribution of blazars to the X-ray background, between 1 and 50 keV, is
approximately constant and of the order of 4-5%. On the basis of an
interpolation between the estimated flux at X-ray and gamma-ray energies we can
expect that the contribution of blazars raises to ~10% at 100 keV, and
continues to increase with energy until it becomes the dominant component at a
few MeV. Finally, we show that a strong dependence of the synchrotron peak
frequency on luminosity, as postulated by the "blazar sequence", is ruled out
by the observational data as it predicts a gamma-ray background above a few GeV
that is far in excess of the observed value.
