  With the coincident detections of electromagnetic radiation together with
gravitational waves (GW170817) or neutrinos (TXS 0506+056), the new era of
multimessenger astrophysics has begun. Of particular interest are the searches
for correlation between the high-energy astrophysical neutrinos detected by the
IceCube Observatory and gamma-ray photons detected by the Fermi Large Area
Telescope (LAT). So far, only sources detected by the LAT have been considered
in correlation with IceCube neutrinos, neglecting any emission from sources too
faint to be resolved individually. Here, we present the first cross-correlation
analysis considering the unresolved gamma-ray background (UGRB) and IceCube
events. We perform a thorough sensitivity study and, given the lack of
identified correlation, we place upper limits on the fraction of the observed
neutrinos that would be produced in proton-proton or proton-gamma interactions
from the population of sources contributing to the UGRB emission and dominating
its spatial anisotropy (aka blazars). Our analysis suggests that, under the
assumption that there is no intrinsic cutoff and/or hardening of the spectrum
above Fermi-LAT energies, and that all gamma-rays from the unresolved blazars
dominating the UGRB fluctuation field are produced by neutral pions from p-p
(p-gamma) interactions, up to 60% (30%) of such population may contribute to
the total neutrino events observed by IceCube. This translates into a O(1%)
maximum contribution to the astrophysical high-energy neutrino flux observed by
IceCube at 100 TeV.
