Which constitutional principle allows the national government to have an impact on state-level civil rights violations under the civil rights act of 1964?

The constitutional principle which might have inspired the Civil Rights Bill of 1964, enacted on July 2nd, of the same year is that of egalitarianism, particularly regarding equal treatment for citizens before the law - 14th Ammendment-, and the right to vote - 15th Ammendment-, because all the provisions in this ruling address the prohibition to discriminate people based on race, color, religion, sex or national origin, specially when regulating labor relations and the right to vote.

The Supreme Court therefore, has the authority to rule over those civil rights violations which took place at a state level, and so they did, for example, in 1974, ruled that the San Francisco school district had violated the students' rights as they did not have separate classes for non-English speakers, mixing students in the same classrooms, because notall the students were ultimately being treated equeally before the law.