The presence of relativistic electrons within the diffuse gas phase of galaxy clusters is now well established, thanks to deep radio observations obtained over the last decade, but their detailed origin remains unclear. Cosmic ray protons are also expected to accumulate during the formation of clusters. They may explain part of the radio signal and would lead to γ-ray emission through hadronic interactions within the thermal gas. Recently, the detection of γ-ray emission has been reported toward the Coma cluster with Fermi-LAT. Assuming that this γ-ray emission arises essentially from pion decay produced in proton-proton collisions within the intracluster medium (ICM), we aim at exploring the implication of this signal on the cosmic ray content of the Coma cluster and comparing it to observations at other wavelengths. We use the MINOT software to build a physical model of the Coma cluster, which includes the thermal target gas, the magnetic field strength, and the cosmic rays, to compute the corresponding expected γ-ray signal. We apply this model to the Fermi-LAT data using a binned likelihood approach, together with constraints from X-ray and Sunyaev-Zel’dovich observations. We also consider contamination from compact sources and the impact of various systematic effects on the results. We confirm that a significant γ-ray signal is observed within the characteristic radius θ500 of the Coma cluster, with a test statistic TS ≃ 27 for our baseline model. The presence of a possible point source (4FGL J1256.9+2736) 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 as it is strongly degenerate with the expected ICM emission, and extended models match the data better. Given the Fermi-LAT angular resolution and the faintness of the signal, it is not possible to strongly constrain the shape of the cosmic ray proton spatial distribution when assuming an ICM origin of the signal, but preference is found in a relatively flat distribution elongated toward the southwest, which, based on data at other wavelengths, matches the spatial distribution of the other cluster components well. Assuming that the whole γ-ray signal is associated with hadronic interactions in the ICM, we constrain the cosmic ray to thermal energy ratio within R500 to XCRp = 1.79−0.30+1.11% and the slope of the energy spectrum of cosmic rays to α = 2.80−0.13+0.67 (XCRp = 1.06−0.22+0.96% and α = 2.58−0.09+1.12 when including both the cluster and 4FGL J1256.9+2736 in our model). 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 (six times lower when including 4FGL J1256.9+2736), so that primary cosmic ray electrons or reacceleration of secondary electrons is necessary to explain the total emission. We constrain the amplitude of the primary to secondary electrons, or the required boost from reacceleration with respect to the steady state hadronic case, depending on the scenario, as a function of radius. Our results confirm that γ-ray emission is detected in the direction of the Coma cluster. Assuming that the emission is due to hadronic interactions in the intracluster gas, they provide the first quantitative measurement of the cosmic ray proton content in a galaxy cluster and its implication for the cosmic ray electron populations.