The Fourteenth Amendment adds prohibitions on state action to the original list of prohibitions on state activity.  It does not allow states to violate the due process rights, equal protection of the law, or immunities and privileges of citizenship of its own citizens.  This is not a blanket statement of dominance by the feds over the states.  And, the branch of the federal government tasked with enforcing those prohibitions is the U.S. Congress, not the federal judiciary.