Yes, the Wasserstein distance between two probability measures can be calculated using the Kantorovich-Rubinstein duality. For reference, check the paper "On the convergence of the empirical measure of eigenvalues of random matrices" by L. Pastur and M. Shcherbina, published in the Journal of Mathematical Physics in 2000. In this paper, the authors calculate the Wasserstein distance between the empirical measure of eigenvalues of GUE and GOE, which can be related to the Wasserstein distance between the underlying probability measures. The result is given in terms of the Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of GUE and GOE.

I hope this helps.

Thank you very much Azita Mayeli. Although I couldn't find the paper you've mentioned I found the book "Eigenvalue Distribution of Large Random Matrices" instead which I gauss is directly related to this subject. There's a section in this book entitled "Convergence of the Normalized Counting Measure of Eigenvalues" in chapter 11, and I'm wondering if this discusses the same situation.