He was appointed an Ensign in the 88th Foot in 1779 and served in Jamaica. He was promoted to Lieutenant in 1780. He returned to England in November 1781 and moved to the 2nd Battalion of the Royals. He transfered to the 71st Foot in Madras in 1786. In 1789 he was appointed Brigade-Major and served under Major-General Sir William Meadows and Governor of India, Lord Cornwallis. He was at the sieges of Bangalore and Sevendroog, the Storming of Tippoo's lines and the surrender of Seringapatam. He was promoted to Captain-Lieutenant 74th Foot on November 1, 1792 and then to Captain in the 73rd on March 7, 1795.
In 1797 he sailed on the projected Manila expedition.
In February 1800 he was appointed Governor of Prince of Wales' Island and invested with the whole civil and military authority and on the 20th of April he arrived and took charge of the government. The Governor-general in council had selected him for the office of lieutenant-governor from his personal knowledge of Sir George's integrity, prudence and firmness and by his lordship's conviction that the services of Sir George Leith would be eminently useful in securing to the company all advantages to be derived from the important settlements (on Prince of Wales' Island). That year Sir George negotiated a further treaty with the Sultan for the cession of a strip of mainland Kedah which was then renamed Province Wellesley after Richard Wellesley, 1st Marquess Wellesley, the Governor-General of India. He continued to govern till 1803 when he was succeeded by Sir Robert Townsend Farquhar. The Prince of Wales' Island, together with Province Wellesley now formed the Settlement of Penang (the island having been locally known as Pulau Pinang, where in the Malay language pulau means island and pinang is the name of a variety of local palm tree).
He received a Majority in the 17th Foot on January 1, 1800 and a Lieutenant-Colonelcy in the 2nd West-India Regiment on June 13, 1805. In 1805, because of his part in quelling a rebellion, he was presented with a set of porcelain by the King of Burma. At this time he had also completed and caused to be published, a book about his experiences on the island up to that point. He continued as Lieutenant-Governor of Penang until 1806 when he received leave to return to England.
He was on the point of embarking for the West Indies, when he was ordered to Ireland as Assistant Assistant Adjutant-General of the British forces there. He attained the Brevet rank of Colonel in 1813, Major-General in 1819 and on the 20th of November of the same year he was appointed Colonel of the 9th Royal Veteran Battalion. He was lmade a Knight-Commander of the Order of the Bath.
In 1805 he wrote and had published the book, "A Short Account of the Settlement, Produce, and Commerce of Prince of Wales Island, in the Straits of Malacca".
He died in Scotland in 1842. Leith Street in Penang is named after him.