The Texas Law, according to Justice Ginsburg, allows testing in limited circumstances:

[The inmate] may show that, at trial, testing either was “not available” or was “available, but not technologically capable of providing probative results.” Art. 64.01(b)(1)(A). Alternatively, he may show that the evidence was not previously tested“through no fault” on his part, and that “the interests of justice” require a postconviction order for testing. Art. 64.01(b)(1)(B).

To grant a motion for postconviction testing, a court must make further findings, prime among them, the movant “would not have been convicted if exculpatory results had been obtained through DNA testing,”and “the [Article 64] request . . . [was] not made to unreasonably delay the execution of sentence or administration of justice.” Art. 64.03(a)(2).

Skinner argued Texas violated his Fourteenth Amendment right to due process. The Court distinguished Skinner's claim by saying;

Because this case was resolved on a motion to dismiss for failure to state a claim, the question below was “not whether [Skinner] will ultimately prevail” on his procedural due process claim... but whether his complaint was sufficient to cross the federal court’s threshold...

It also noted that Skinner was not challenging the prosecutor's conduct or the lower court's application of the law, but the law itself. The Court said it was ruling only on the jurisdictional issue and on whether Skinner had a claim:

We take up here only the questions whether there is federal-court subject matter jurisdiction over Skinner’s complaint, and whether the claim he presses is cognizable under §1983.

....

Skinner does not challenge the adverse CCA decisions themselves; instead, he targets as unconstitutional the Texas statute they authoritatively construed. .... a state-court decision is not reviewable by lower federal courts, but a statute or rule governing the decision may be challenged in a federal action. Skinner’s federal case falls within the latter category. There was, therefore, no lack of subject-matter jurisdiction over Skinner’s federal suit.

On using the civil rights statute:

Skinner has properly invoked §1983. Success in his suit for DNA testing would not “necessarily imply” the invalidity of his conviction. While test results might prove exculpatory, that outcome is hardly inevitable; as earlier observed, see supra, at 2, results might prove inconclusive or they might further incriminate Skinner.

His lawyer has issued a statement, which includes the following:

"The Court's action corrects the Fifth Circuit’s fundamental misunderstanding of this important principle. As Justice Ginsberg states in her majority opinion, there is no reason to fear that lawsuits like Mr. Skinner's will overwhelm the federal courts. The high court’s ruling will simply make it possible for Mr. Skinner to vindicate his due process rights in federal court, a right long enjoyed by prisoners in other parts of the country.

"We look forward to making our case in federal court that Texas's inexplicable refusal to grant Mr. Skinner access to evidence for DNA testing is fundamentally unfair and cannot stand."