It is one of several conservative decisions made by the court of late, which has swung to the right thanks to a series of high-profile Republican appointments made by President George Bush.
His pals Chief Justice John Roberts and Justice Samuel Alito made headlines with their right-wing stances on abortion.
Responding to the court ruling, Mrs Clinton has said it would "erode core constitutional values." She was joined by fellow Democratic runner Barack Obama, who dismissed the judgment as "deeply wrong-headed".
The case centres on the voluntary racial diversity programmes adopted in Seattle and Louisville in Kentucky, aimed at preserving the racial balance in schools.
The first case was brought by the mother of a pupil who had been denied a transfer to a different nursery because his existing nursery wanted to preserve their quota of white pupils. The second was brought by Seattle parents wishing to fight the city's high school assignments plans.
Explaining the court's decision, Chief Justice John Roberts said the state should be colour-blind. In the UK, the idea of school race quotas has been mooted by the Conservatives, but has not yet been given serious consideration.