We experimentally and theoretically demonstrate that nonlinear spin-wave interactions suppress the hybrid magnon-photon quasiparticle or "magnon polariton" in microwave spectra of a yttrium iron garnet film detected by an on-chip split-ring resonator. We observe a strong coupling between the Kittel and microwave cavity modes in terms of an avoided crossing as a function of magnetic fields at low microwave input powers, but a complete closing of the gap at high powers. The experimental results are well explained by a theoretical model including the three-magnon decay of the Kittel magnon into spin waves. The gap closure originates from the saturation of the ferromagnetic resonance above the Suhl instability threshold by a coherent backreaction from the spin waves.