Maybe, but probably not.
An employer is fully within its right to discipline you for insubordination. If you spoke to him "extremely harshly" then the employer could well have terminated you without warning or prior misconduct. Apparently, he chose to simply suspend you. Given that you admit to have engaged in such conduct, it is likely the employer could well defend its disciplinary conduct.
To prove race discrimination, you would need to prove that the reason you were disciplined was because you were Caucasian and not another race. Even if you could prove that your race was one of the reasons why you were disciplined and not another employee who engaged in the same conduct, unless you could prove that the insubordination was not a motivating factor for your discipline, you would likely get no damages for the discriminatory discipline. New California Supreme Court case law says that where the employer can establish that a motivating factor was a non-discriminatory act, then the employee cannot recover damages.
Ultimately, at this point, a one day suspension is not enough to even right about this. However, keep a log of all of the comments and unfair treatment you are facing based on your race. As it accumulates, there is a greater chance of proving a racially hostile work environment, for which there might be damages.
Good luck to you.
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Disparate treatment in the employment context is typically found when "“the employer treats some people less favorably than others because of their race, color, religion, sex, or national origin.” Further what you have described may be a violation of a progressive discipline policy (counseling or a write up before a suspension) as well as an example of disparate treatment (aka "discrimination") if similarly situated non-white employees were not suspended for similar conduct. You are also protected for "protesting" such unlawful conduct (from a wrongful termination for example) if you pursue this matter. The use of a racial epithet (the "white girl") is equally inappropriate and disturbing. You may wish to contact the DFEH or a civil attorney if you feel this matter may escalate. The DFEH is a necessary step (whether you avail yourself of the administrative route or if you decide to proceed to a "Right to Sue") before pursuing the employer. Many employment attorneys offer a free and confidential consultation as this may be a lot to take in at first.
I have attached a helpful link from the DFEH for your convenience. I am sorry you have experienced this.