During the Arthur administration, the LDS Church was under government pressure to stop the practice of polygamy in the Utah Territory. In 1882, the president signed the Edmunds Act, which made polygamy a federal crime and barred polygamists from public office and the right to vote.

Arthur made appointments to fill two vacancies on the Supreme Court. The first vacancy arose in July 1881 with the death of Associate Justice Nathan Clifford, a Democrat who had been a member of the Court since before the Civil War. Arthur nominated Horace Gray to replace him, and the nomination was easily confirmed. The second vacancy occurred when Associate Justice Ward Hunt retired in January 1882. Arthur first nominated his old political boss, Roscoe Conkling, but after the Senate confirmed the nomination, Conkling declined it, marking the last time a confirmed nominee declined an appointment. Arthur later nominated Samuel Blatchford, and his nomination was approved by the Senate within two weeks.

Shortly after becoming president, Arthur was diagnosed with Bright's disease, a kidney ailment now referred to as nephritis. He attempted to keep his condition private, but by 1883 rumors of his illness began to circulate. In an attempt to rejuvenate his health, he visited Yellowstone National Park. Reporters accompanied the presidential party, helping to publicize the new National Park system, and Arthur returned to Washington refreshed after two months of travel.

After spending the summer of 1886 in Connecticut, he returned home where he became seriously ill, and on November 16, ordered nearly all of his papers, both personal and official, burned. The next morning, Arthur suffered a cerebral hemorrhage and never regained consciousness. He died the following day, November 18, at the age of 57.