Son of Alexander Stewart, 6th Earl of Galloway and Catherine Cochrane. Brother of John Stewart and Catherine Stewart. Married (1782) to Georgiana Isabella D'Aguilar. A Captain in 1762 he later reached the high rank of Admiral. In the late 1780s, Stewart had acquired enough land to have become one of the largest landowners in Ayrshire. He was also sometime Receiver General of the Land Tax for Scotland.
His paternal grandmother was Catherine Montgomerie (d. 1757), sister of the 10th and 11th Earls of Eglinton.
Boswell went with Stewart and Stewart's brother-in-law James Murray to a sermon in St. Johns Church in London on December 19, 1762.
In 1789 Boswell was involved, with Stewart and Thomas Miller, in a dispute over the relocation of a tollbar in the parish of Muirkirk.1 Boswell and Stewart also met, coincidentally, in Italy and Toulon during Boswell's Grand Tour in 1765. Stewart seems to have been in active service in the Mediterranean as a naval captain at the time, and when they met in Florence, he warned Boswell about going to Corsica.