Prior to the 14th Amendment, you could make a good case that the protections of the Constitution only applied to the federal government. However, the first clause of the 14th Amendment changed that (emphasis added):

1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

Notice that this sentence is specifically aimed at the states. This clause opens the federal courts for jurisdiction in what previously were considered 'states rights' issues. Given the history of Jim Crow laws, there was some justification for the equal protection clause.