Recent remote sensing measurements and numerical studies have shown that surface gravity waves interact strongly with small-scale open ocean currents. Through these interactions, the significant wave height, the wave frequency, and the wave direction are modified. In the present paper, we investigate the interactions of surface gravity waves with a large and isolated realistic cyclonic eddy. This eddy is subject to instabilities, leading to the generation of specific features at both the mesoscale and submesoscale ranges. We use the WAVEWATCH III numerical framework to force surface gravity waves in the eddy before and after its destabilization. In the wave simulations the source terms are deactivated, and waves are initialized with different wave intrinsic frequencies. The study of these simulations illustrates how waves respond to the numerous kinds of instabilities in the large cyclonic eddy from a few hundred to a few tens of kilometres. Our findings show that the spatial variability of the wave direction, the mean period, and the significant wave height is very sensitive to the presence of submesoscale structures resulting from the eddy destabilization. The intrinsic frequency of the incident waves is key in the change of the wave direction resulting from the current-induced refraction and in the location, from the boundary where waves are generated, of the maximum values of significant wave height. However, for a given current forcing, the maximum values of the significant wave height are similar regardless of the frequency of the incident waves. In this idealized study it has been shown that the spatial gradients of wave parameters are sharper for simulations forced with the destabilized eddy. Because the signature of currents on waves encodes important information of currents, our findings suggest that the vertical vorticity of the current could be statistically estimated from the significant wave height gradients down to a very fine spatial scale. Furthermore, this paper shows the necessity to include currents in parametric models of sea-state bias; using a coarse-resolution eddy field may severely underestimate the sea-state-induced noise in radar altimeter measurements.