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]