Bolling v. Sharpe is, I think, one of the most interesting cases that came up before the Supreme Court. While logic and reason might suggest that educational segregation is unconstitutional both in the states and the District of Columbia, the plain text of the Constitution actually provides the opposite.
Of course, the equal protection clause cannot apply to the federal government. The Fourteenth Amendment is quite clear that it applies only to the states. Chief Justice Earl Warren admits it in his opinion, writing, "The Fifth Amendment, which is applicable in the District of Columbia, does not contain an equal protection clause as does the Fourteenth Amendment which applies only to the states."
Thus, the only possible justification for this ruling is the due process clause. But the due process clause only prevents the government from depriving an individual of life, liberty, or property without due process of law. Obviously, segregation in public schools does not deprive an individual of life or property. Chief Justice Warren argued that the term liberty "is not confined to mere freedom from bodily restraint." While this is perfectly true, the protection of "liberty" cannot be reasonably held to extend to public education. Indeed, the provision of a government service is a far cry from "liberty"; the former is only a political matter provided at the discretion of the government, whereas the latter is enshrined and protected by the Constitution which the government is required to respect. Although I have always argued for a broad interpretation of the word "liberty," I cannot accept that the term includes participation in government programs.*
But, even if we accept the absurd assertion that the protection of liberty includes a right to government services, public segregation in the District of Columbia would not be unconstitutional under the due process clause. A law cannot violate the due process clause, because it is the legislature which defines "due process of law" in the first place. Due process protections are purely procedural, not "substantive"; they are defined by the law, so the law cannot violate them. (Only the executive or the judiciary can do so.)
Thus, the due process clause cannot be invoked here for two reasons: firstly, the clause protects "liberty," not participation in government programs; secondly, the protections are purely procedural, nothing further.
Accordingly, I would have to reluctantly conclude that Bolling was wrongly decided. The District of Columbia law in question was, undoubtedly, an "uncommonly silly" one (to repeat the words of Potter Stewart). Indeed, the law was unjust, oppressive, and reprehensible. But it was not unconstitutional.
*Note: If we accept that "liberty" includes a right to participate in a government program, as well as the doctrine of substantive due process, then we reach a very absurd interpretation. It could be argued that the government could not pass a law abolishing a program, because doing so would "deprive an individual of liberty without substantive due process." Clearly, therefore, the assumption that liberty encompasses government services would be an absurdity