I'm not sure there is such a thing as "authentic" Mediterranean cuisine because it's usually a mix of Greek, Turkish, Moroccan, and even Italian and French.
In the Union Square area, Barbounia, on the corner of Park Av. S. & 20th St., is considered a Mediterranean restaurant. We've had several good meals there although I just learned that there has recently been a chef change, and so the menu, while remaining "Mediterranean," has been revamped since our last time there. It's in a huge, soaring space with very eye-catching decor.