We investigate the IceCube detection potential of very high energy neutrinos from blazars, for different classes of hadronic models, taking into account the limits imposed on the neutrino flux by the recent Fermi telescope observations. Assuming the observed gamma-ray emission is produced by the decay of neutral pions from proton-proton interactions, the measurement of the time-averaged spectral characteristics of blazars in the GeV energy band imposes upper limits on the time-averaged neutrino flux. Comparing these upper limits to the 5 sigma discovery threshold of IceCube for different neutrino spectra and different source locations in the sky, we find that several BL Lacs with hard spectra in the GeV band are within the detection potential of IceCube. If the gamma-ray emission is dominated by the neutral pion decay flux, none of the flat-spectrum radio quasars are detectable with IceCube. If the primary high energy proton spectrum is very hard and/or neutrinos are produced in proton photon, rather than proton-proton reactions, the upper limit on the neutrino flux imposed by the measured gamma-ray spectra is relaxed and gamma-ray observations impose only lower bounds on the neutrino flux. We investigate whether these lower bounds guarantee the detection of blazars with very hard neutrino spectra (spectral index Gamma(nu) similar to 1), expected in the latter type model. We show that all the hadronic models of activity of blazars are falsifiable with IceCube. Furthermore, we show that models with gamma-ray emission produced by the decay of neutral pions from proton-proton interactions can be readily distinguished from the models based on proton-gamma interactions and/or models predicting very hard high energy proton spectra via a study of the distribution of spectral indices of gamma-ray spectra of sources detected with IceCube.