  A fraction of merging galaxy clusters host diffuse radio emission in their
central region, termed as a giant radio halo (GRH). The most promising
mechanism of GRHs is the re-acceleration of non-thermal electrons and positrons
by merger-induced turbulence. However, the origin of these seed leptons has
been under debate, and either protons or electrons can be primarily-accelerated
particles. In this work, we demonstrate that neutrinos can be used as a probe
of physical processes in galaxy clusters, and discuss possible constraints on
the amount of relativistic protons in the intra-cluster medium with the
existing upper limits by IceCube. We calculate radio and neutrino emission from
massive ($>10^{14}M_\odot$) galaxy clusters, using the cluster population model
of Nishiwaki & Asano (2022). This model is compatible with the observed
statistics of GRHs, and we find that the contribution of GRHs to the isotropic
radio background observed with the ARCADE-2 experiment should be subdominant.
Our fiducial model predicts the all-sky neutrino flux that is consistent with
IceCube's upper limit from the stacking analysis. We also show that the
neutrino upper limit gives meaningful constraints on the parameter space of the
re-acceleration model, such as the electron-to-proton ratio of primary
cosmic-rays and the magnetic field, and in particular the secondary scenario,
where the seed electrons mostly originate from inelastic $pp$ collisions, can
be constrained even in the presence of re-acceleration.
