Where does the race part come into play here? Mexican is a nationality.
Then she made an ethnic slur and where's the denial for that?
While that is true, it's also used as a quasi racial designation. It's what Texas Hispanics were called before the term "Hispanic" came into use. Well "Latino/Latina" were in there somewhere too. The terms are of course more ethnic than racial.
However I did have a Hispanic lady in far west San Antonio ask me if my wife was "white" too, which she is, when I told her that my wife could speak Spanish fluently, with a mixed Cuban/Spainish accent. :). However the lady's skin was every bit as white as my wife's, the difference being in hair and eye color, not skin color.