  Gamma-ray bursts (GRBs) are proposed as candidate sources of ultra-high
energy cosmic rays (UHECRs). We study the possibility that the PeV neutrinos
recently observed by IceCube are produced by GRB cosmic rays interacting with
the interstellar gas in the host galaxies. By studying the relation between the
X-ray absorption column density N_H and the surface star-formation rate of GRB
host galaxies, we find that N_H is a good indicator of the surface gas density
of the host galaxies. Then we are able to calculate the neutrino production
efficiency of CRs for GRBs with known N_H. We collect a sample of GRBs that
have both measurements of N_H and accurate gamma-ray fluence, and attempt to
calculate the accumulated neutrino flux based on the current knowledge about
GRBs and their host galaxies. When the CR intensity produced by GRBs is
normalized with the observed UHECR flux above $10^{19}{\rm eV}$, the
accumulated neutrino flux at PeV energies is estimated to be about
$(0.3\pm0.2)\times10^{-8} \rm{GeV\ cm^{-2}\ s^{-1}\ sr^{-1}} $ (per flavor)
under the assumption that GRB energy production rate follows the cosmic
star-formation rate and the favorable assumption about the CR diffusion
coefficient. This flux is insufficient to account for the IceCube observations,
but the estimate suffers from some assumptions in the calculation and thus we
can not rule out this scenario at present.
