  In this work, starting from 21 months of data from the Fermi-Large Area
Telescope, we derive maps of the residual isotropic gamma-ray emission, a
relevant fraction of which is expected to be contributed by the extragalactic
diffuse gamma-ray background. We compute the angular two-point auto-correlation
function of the residual Fermi-LAT maps at energies E>1GeV, E>3GeV and E>30GeV
well above the Galactic plane and find no significant correlation signal. This
is, indeed, what is expected if the EGB were contributed by BL Lacertae, Flat
Spectrum Radio Quasars or star-forming galaxies, since, in this case, the
predicted signal is very weak. Then, we search for the Integrated Sachs-Wolfe
signature by cross-correlating the Fermi-LAT maps with the WMAP7-Cosmic
Microwave Background map. We find a cross-correlation consistent with zero,
even though the expected signal is larger than that of the EGB
auto-correlation. Finally, in an attempt to constrain the nature of the
gamma-ray background we cross-correlate the Fermi-LAT maps with the angular
distributions of objects that may contribute to the EGB: QSOs in the SDSS-DR6
catalog, NVSS galaxies, 2MASS galaxies and LRG in the SDSS catalog. The
cross-correlation is always consistent with zero, in agreement with theoretical
expectations, but we find (with low statistical significance) some interesting
features that may indicate that some specific classes of objects contribute to
the EGB. A chi2 analysis confirms that the correlation properties of the
21-month data do not provide strong constraints of the EGB origin. However, the
results suggest that the situation will significantly improve with the 5- and
10-year Fermi-LAT data. The future EGB analysis will then allow placing
significant constraints on the nature of the EGB and might provide in addition
a detection of the ISW signal. (Abridged)
