A substantial number of processes have been suggested as possible contributors to the extragalactic $gamma$-ray background (EGRB). Yet another contribution to this background will be emission produced in hadronic interactions of cosmic-ray protons with the cluster thermal gas; this class of cosmic rays (CRs) has been shown to be responsible for the EUV emission in the Coma Cluster of galaxies. In this paper we assume the CRs in the Coma Cluster is prototypic of all clusters and derive the contribution to the EGRB from all clusters over time. We examine two different possibilities for the scaling of the CR flux with cluster size: the number density of the CRs scale with the number density of the thermal plasma, and alternatively, the energy density of the CRs scale with the energy density of the plasma. We find that in all scenarios the EGRB produced by this process is sufficiently low that it will not be observable in comparison with other mechanisms that are likely to produce an EGRB.