  The presence of relativistic electrons within the diffuse gas phase of galaxy
clusters is now well established, but their detailed origin remains unclear.
Cosmic ray protons are also expected to accumulate during the formation of
clusters and would lead to gamma-ray emission through hadronic interactions
within the thermal gas. Recently, the detection of gamma-ray emission has been
reported toward the Coma cluster with Fermi-LAT. Assuming that this gamma-ray
emission arises from hadronic interactions in the ICM, we aim at exploring the
implication of this signal on the cosmic ray content of the Coma cluster. We
use the MINOT software to build a physical model of the cluster and apply it to
the Fermi-LAT data. We also consider contamination from compact sources and the
impact of various systematic effects. We confirm that a significant gamma-ray
signal is observed within the characteristic radius $\theta_{500}$ of the Coma
cluster, with a test statistic TS~27 for our baseline model. The presence of a
possible point source may account for most of the observed signal. However,
this source could also correspond to the peak of the diffuse emission of the
cluster itself and extended models match the data better. We constrain the
cosmic ray to thermal energy ratio within $R_{500}$ to $X_{\rm
CRp}=1.79^{+1.11}_{-0.30}$\% and the slope of the energy spectrum of cosmic
rays to $\alpha=2.80^{+0.67}_{-0.13}$. Finally, we compute the synchrotron
emission associated with the secondary electrons produced in hadronic
interactions assuming steady state. This emission is about four times lower
than the overall observed radio signal, so that primary cosmic ray electrons or
reacceleration of secondary electrons is necessary to explain the total
emission. Assuming an hadronic origin of the signal, our results provide the
first quantitative measurement of the cosmic ray proton content in a
cluster.[Abridged]
