On September 26, 2013, Governor Perry appointed Brown to the Texas Supreme Court to fill the seat of Nathan Hecht, who was elevated to chief justice.

[6] His service on the Texas Supreme Court ended on September 4, 2019, when he was commissioned as a federal district judge.

[15] In January 2022, Brown enjoined enforcement of a COVID-19 vaccine mandate for federal employees that President Biden's administration had implemented.

[16] Noting his belief that people should get vaccinated against COVID-19, Brown explained that the case turned on the question of "whether the President can, with the stroke of a pen and without the input of Congress, require millions of federal employees to undergo a medical procedure as a condition of their employment.

"[17] In April 2022, two judges on a three-judge panel of the United States Court of Appeals for the Fifth Circuit vacated Brown's ruling,[18] but in June a majority of active Fifth Circuit judges voted to rehear the cause en banc, thereby vacating the April panel opinion.

[19] In March 2023, the Fifth Circuit, sitting en banc, affirmed Brown's nationwide injunction; Judge Andrew Oldham wrote the opinion for a ten-member majority.

[20] In October 2023, in what was the first Voting Rights Act case decided since Allen v Milligan, Brown ruled that a newly adopted map for local elections in Galveston County, Texas, violated § 2 of the VRA by eliminating the county's one majority-minority precinct and thereby making less likely the election of a candidate preferred by a majority of Black and Latino voters.

But "the members of th[e] panel agree[d] that th[e] court’s precedent permitting aggregation" of "distinct minority groups like blacks and Hispanics . . .

"[22] Later that month, a majority of active Fifth Circuit judges voted to rehear the case en banc.