The Courts have ruled that the Constitution gives the Federal government the authority to Establish Justice via—for example—ensuring Equal Protection (and Opportunity) under the law, among other things. And "discrimination based on race, color, religion, sex, or national origin" has been deemed a violation of Equal Protection.
Inasmuch as there is any law—at any level of government—which establishes (or permits) a violation of Equal Protection, that's where such Federal authority would originate. In such cases, the Judicial Branch would determine whether such law violated Equal Protection and was therefore unconstitutional.
Those Court rulings have no constitutional basis. And the Constitution trumps any unconstitutional Court ruling (U.S. Const, art. VI, cl. 2).
Nowhere does the Constitution as written and originally understood and intended grant the feds the sweeping powers they have assumed to prohibit discrimination based on race, color, religion, sex, or national origin. The Founders never contemplated such powers and there is NO evidence that the ratifiers of the 14A contemplated such radical shift of power to the federal government. And the intent of the ratifiers is decisive.
Again, in The Slaughterhouse Cases of 1873, a few years after the ratification of the 14th Amendment, the Supreme Court confirmed that the 14A, like the 13A and the 15A, were Post-Civil-War Reconstruction Amendments ratified and limited to instate ex-slaves as full citizens, and not meant to grant new sweeping powers to the feds. The so-called "Incorporation Doctrine" has been weighed and found to be counterfeit having no genuine constitutional basis.