  The IceCube high-energy neutrino telescope has been collecting data since
2006. Conversely, hundreds of Gamma Ray Bursts (GRBs) have been detected by the
GBM on board Fermi, since its launch in 2008. So far no neutrino event has been
associated with a GRB, despite many models predicting the generation of high
energy neutrinos through GRB photon interaction with PeV protons in the GRB
jet. We use the non-detection of neutrinos to constrain the hadronic content of
GRB jets independent of jet model parameters. Assuming a generic particle
spectrum of $E^{-\alpha}$ with $\alpha = 2$, we find that the ratio of the
energy carried by pions to that in electrons has to be small $f_\pi / f_e
\lesssim 0.24$ at 95\% confidence level. A distribution of spectral slopes can
lower $f_\pi / f_e$ by orders of magnitude. Another limit, independent of
neutrinos, is obtained if one ascribes the measured Fermi/LAT GeV gamma-ray
emission to pair-photon cascades of high-energy photons resulting from (the
same photon-hadronic interactions and subsequent) neutral pion decays. Based on
the generally observed MeV to GeV GRB fluence ratio of $\approx 10$, we show
that $f_\pi / f_e \lesssim 0.3$. In some bursts, where this ratio is as low as
unity, $f_\pi / f_e \lesssim 0.03$. These findings add to the mounting doubts
regarding the presence of PeV protons in GRB jets.
