There were two surgeons by the name of James Hall -
(1) James Hall. Date of Seniority Royal Navy 2 August 1808
(2) James Hall. Date of Seniority Royal Navy 28 August 1809
From the Australian Dictionary of Biography Online - James Hall, naval surgeon, was born on 17 September 1784 at New Inn Yard, Shoreditch, London, son of Joseph Hall and his wife, Mary Shaw.
James was a second surgeon at the naval hospital at Corfu, Greece, in 1807. At the risk of his life he warned a British sloop that the French had occupied Corfu, thus saving her from seizure, and gave the commander of H.M.S. Weazle information which enabled him to sink three privateers and capture a gunboat with eight transports under convoy, loaded with French troops bound for Corfu.
He was appointed an assistant surgeon in the navy in August 1809 and surgeon in September 1817. 
James Hall was employed as Surgeon-Superintendent on four convict ships to Australia:
Georgiana to Tasmania in 1833
departed Portsmouth on 3 May 1820 with 179 male prisoners and arrived in Port Jackson on 22 September 1820. One prisoner died on the passage out.
James Hall kept a Medical Journal from 6 April to 29 September 1820. He signed the Journal of the Agamemnon J. Hall, Surgeon, formerly of the Imperial Russian Navy.
Mary Anne 1822
The Mary Anne
departed Portsmouth on 22 December 1821 with 108 female prisoners. They sailed via Rio de Janeiro and Van Diemen's Land where forty-five female convicts were landed before the ship proceeded to Sydney where the remaining 62 women were disembarked. James Hall kept a Medical Journal from 27 October 1821 to 25 May 1822
The case of death that occurred was one that might have been expected so to terminate as the woman had not menstruated for several years and had long laboured under chronic visceral diseases. The immediate cause of death seemed to be a cancerous affection of the stomach.
On the whole I must observe that the women and children enjoyed a high state of health during the voyage owing to a rigid system of cleanliness and good discipline having been maintained.
departed the Downs on 6 September 1823 with 89 female prisoners.
James Hall was assaulted by female prisoners in December 1823 when the Brothers, had been at sea for only a week. Later in a series of court actions between the officers of the ship, James Hall's lack of tact was seen as the primary cause of the trouble. Some of the ship's crew, passengers and convicts later gave evidence at an enquiry. James Hall kept a Medical Journal from 16 October 1823 to 15 May 1824
In the early part of the voyage many women suffered sea sickness. There were also cases of ophthalmia and catarrh and at least two births. Free passenger Mrs. Butler gave birth to a child on 17 February and Bridget Hanning was 44 years old when she gave birth to her eleventh child on the 17th March. Two days later the baby was found dead in its mother's arms. Other deaths that occurred included Mary Partridge who was 21 years old and pined for her mother she had left behind; and the six year old son of Mrs. Butler. Mr Butler later died as well. Altogether the surgeon's journal recorded five deaths, none of whom were convicts.
The Georgiana departed Portsmouth with 184 male prisoners on 16 October 1832 and arrived in Van Diemen's Land on 1 February 1833. James Hall kept a Medical Journal from 8 September 1832 to 7 February 1833.
James Hall M.D. (28 August 1809) was on the List of Retired Surgeons of the Royal Navy in 1841.
Notes and Links
1). Read an in depth account of James Hall's life and career select King-Hall Connections.
2). Sea Saga - L. King Hall
- Internet Archive
 'Hall, James (1784–1869)', Australian Dictionary of Biography, National Centre of Biography, Australian National University