Hosack was the family doctor of Alexander Hamilton and his family, and is perhaps best known as the doctor present during Hamilton's duel with Aaron Burr in Weehawken, New Jersey on July 11, 1804. Hosack, who was a friend to both Hamilton and Burr, treated Hamilton after Burr fired, and accompanied him across the river to the dockside home of William Bayard Jr., where Hamilton died the next day. Hosack had also treated Hamilton's son Philip after he was fatally shot by George Eacker in a duel at the same location in 1801. At Alexander Hamilton's funeral, Hosack was one of the pallbearers.