Through-wall radars are researched and developed for the detection, localization, and tracking of human activities in indoor environments. Electromagnetic wave propagation through walls introduces refraction, attenuation, multipath, and ghost targets in the radar signatures. Estimation of wall characteristics (dielectric profile and thickness) can enable wall effects to be deconvolved from through-wall radar signatures. We propose to use generative adversarial networks (GAN) to estimate wall characteristics from narrowband scattered electric fields on the same side of the wall as the transmitter. We demonstrate that the GANs, consisting of two neural networks configured in an adversarial manner, are capable of solving the highly nonlinear regression problem with limited training data to estimate the dielectric profile and thickness of actual walls up to 95\% accuracy based on training with simulated data generated from full-wave solvers.