  Context: Hadronic models, involving matter (proton or nuclei) acceleration in
blazar jets, imply high energy photon and neutrino emissions due to
interactions of high-energy protons with matter and/or radiation in the source
environment. Aims: This paper shows that the sensitivity of the IceCube
neutrino telescope in its 40-string configuration (IC-40) is already at the
level of constraining the parameter space of purely hadronic scenarios of
activity of blazars. Methods: Assuming that the entire source power originates
from hadronic interactions, and assuming that the models describe the data, we
estimate the expected neutrino flux from blazars based on the observed
gamma-ray flux by Fermi, simultaneously with IC-40 observations. We consider
two cases separately to keep the number of constrainable parameters at an
acceptable level: proton-proton or proton-gamma interactions are dominant.
Comparing the IC-40 sensitivity to the neutrino flux expected from some of the
brightest blazars, we constrain model parameters characterizing the parent
high-energy proton spectrum. Results: We find that when pp interactions
dominate, some constraints on the primary proton spectrum can be imposed. For
instance, for the tightest constrained source 3C 454.3, the very high energy
part of the spectra of blazars is constrained to be harder than E^-2 with
cut-off energies in the range of Ecut >10^18 eV. When interactions of
high-energy protons on soft photon fields dominate, we can find similarly tight
constraints on the proton spectrum parameters. [abridged]
