We examine the cosmic-ray protons (CRp) accelerated at collisionless shocks in galaxy clusters using cosmological structure formation simulations. We find that in the intracluster medium (ICM) within the virial radius of simulated clusters, only $sim7$% of shock kinetic energy flux is dissipated by the shocks that are expected to accelerate CRp, that is, supercritical, quasi-parallel ($Q_parallel$) shocks with sonic Mach number $M_sge2.25$. The rest is dissipated at subcritical shocks and quasi-perpendicular shocks, both of which may not accelerate CRp. Adopting the diffusive shock acceleration (DSA) model recently presented in Ryu et al. (2019), we quantify the DSA of CRp in simulated clusters. The average fraction of the shock kinetic energy transferred to CRp via DSA is assessed at $sim(1-2)times10^{-4}$. We also examine the energization of CRp through reacceleration using a model based on the test-particle solution. Assuming that the ICM plasma passes through shocks three times on average through the history of the universe and that CRp are reaccelerated only at supercritical $Q_parallel$-shocks, the CRp spectrum flattens by $sim0.05-0.1$ in slope and the total amount of CRp energy increases by $sim40-80$% from reacceleration. We then estimate diffuse $gamma$-ray and neutrino emissions, resulting from inelastic collisions between CRp and thermal protons. The predicted $gamma$-ray emissions from simulated clusters lie mostly below the upper limits set by Fermi-LAT for observed clusters. The neutrino fluxes towards nearby clusters would be $lesssim10^{-4}$ of the IceCube flux at $E_{ u}=1$ PeV and $lesssim10^{-6}$ of the atmospheric neutrino flux in the energy range of $E_{ u}leq1$ TeV.
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