Variously in records as J. Caldwell, Josiah Caldwell and Joseph Caldwell, he was born in Co. Tyrone, Ireland c. 1818.
Joseph Caldwell is entered in the Medical Register 1865 with the following qualifications :
Member Royal College Surgeons England 1841.
Royal Navy Service
Joseph Caldwell was promoted to Surgeon in the Royal Navy in 1842. 
He was appointed to the Mariner sloop in the East Indies on 12 February 1847 
His first appointment as Surgeon-Superintendent on a convict ship was to the Lady Kennaway sailing London to Van Diemen's Land in 1851. While there he attended the Cascades Female House of Correction in September 1851 and gave a joint review of the institution.....We the undersigned have this day visited the Female House of correction and we do express much pleasure in stating that a better conducted establishment has seldom fallen to our lot to have witnessed in any part of the world that we have visited - signed J.M.B. Jones M.D., Joseph Caldwell Surgeon R.N, William Arthur, SurgeonR.N. 
He was appointed to the following Convict Ships in 1850's:
To the Rodney in 1853  The Rodney sailed from Cork to Van Diemen's Land arriving 12 February 1853. He kept a Medical Journal from 19 October 1852 to 18 February 1853.;
To Sea Park from London 1 January 1854 to Fremantle arriving 5 April 1854;  Medical and surgical journal of H.M hired convict ship Sea Park for 1 December 1853 to 21 April 1854 by Joseph Caldwell, Surgeon, during which time the said ship was employed in conveying convicts from England to Fremantle, Western Australia. - National Archives Reference: ADM 101/253/1E
To Stag from London 5 February 1855 to Fremantle arriving 23 May 1855
To Clara from London 19 March 1857 to Western Australia arriving 3 July 1857.
Surgeon H.M.S. Caesar
In 1858 he was engaged as surgeon on the 400 horse power screw steam ship Caesar in the Mediterranean. 
He was still on H.M.S. Caesar on the night of 7 April 1861 when the Census took place. He is single, age 43 and gives his place of birth as Co. Tyrone Ireland.
Court-Martial of Joseph Caldwell
In 1862 Joseph Caldwell was court-martialled for using provoking language and sentenced to be dismissed from His Majesty's service.......
A Court Martial assembled last week at Portsmouth to try Joseph Caldwell, Esq., Staff-Surgeon of her Majesty’s ship Caesar, at Portsmouth, on two charges. Capt. Mason; of the Caear, prosecuted ; and Mr. Wallis appeared as prisoner’s ' friend.'
The first charge was “ That the said Joseph Caldwell, being in actual service and full-pay in her Majesty’s Fleet, and doing duty as staff-surgeon on board ship Caesar, did, on or about five p.m.. on Dec. 20th, 1861, use provoking language tending to made a quarrel and disturbance by twice saying in the ward-room of the said ship, many of the officers being present, ‘ That any member of the mess who said, or who implied, that he wished to get out of any bet that he had made, was a liar and a blackguard, and he would prove it,’ or words to that effect.
Secondly, that prisoner did at the same time and place use provoking language to Lieut. J. C. Pattison, of the same ship, by saying to him, ‘ You lie, Sir; you lie! such lauguage tending to cause a quarrel and disturbance.”
The affair, it appeared from evidence, arose out of a bet having been made between prisoner and one of the lieutenants in reference to the time the ship would take to reach the gut of Gibraltar. An apology (produced by Lieut. Pattison) was tendered to the members of the mess by the prisoner, and was accepted by sixteen of them—two only refusing to acknowledge it. It also appeared that Lieut. Pattison wrote to Capt. Mason asking him to withdraw the charge, as he was satisfied with the apology.
The Court, after deliberating about an hour and a half, found the prisoner guilty on both charges, and he was sentenced to be dismissed her Majesty’s Service.
We cannot but think that the sentence is hard and oppressive. It is certain that under any provocation no man has the right to give the lie to another under insulting conditions; but it is clear that there was provocation here, and it is a stern measure of justice indeed that for an angry word spoken in a mess-room, under the smart of painful insinuations, a gentleman should forfeit the benefit of his long previous services, be dismissed from the Royal service, and deprived of his honourable position and livelihood.
In this case the most ample apologies were offered and accepted, and so the wrong wiped out and condoned. How often has such an angry word been spoken in mess-room and club, and an ample apology has always been held to atone for the offence and heal the wound of the most sensitive honour. If it were necessary to inflict some punishment, then surely some minor sentence, such as being placed at the bottom of the list of surgeons, and consequent loss of promotion, might have sufficed. We repeat that the sentence is oppressive, and therefore unjust. It will grate harshly on the feelings of the civil profession, to whom it must convey an ominous warning. 
Joseph Caldwell played a minor part in one of Australia's more infamous episodes. He was a witness at the trial of William James O'Farrell for the attempted murder of Prince Alfred in 1868.
Joseph Caldwell, surgeon R.N. deposed: I arrived in Sydney a few months ago; know the prisoner; have dined with him daily; on one occasion we read the news that a vessel was observed passing the port of Adelaide supposed to be a Fenian cruiser; he became excited; I never knew him to be incoherent on any one subject; I thought him a little cler; we spoke on all subjects; I long suspected he was a Fenian; few were present during these conversations; this was at the Clarendon Hotel
Cross examined: I am still a surgeon in the Royal Navy; it was in 1862 that I was last on board with passengers; I became acquainted with the prisoner the first day I came to Sydney because I told the cabman to set me down at the first hotel in Geogrge Street and he set me down at the Clarendon; I have been 21 years in active service in the Navy; I was twenty one years of age when I first went in the Navy as assistant surgeon; It is a very silly question to ask me if I had received an education for the treatment of the insane before going on board; I have since had the ordinary experience of other medical practitioners. 
Joseph Caldwell R.N., M.R.C.S.E., died at Portsmouth on 27 April 1886 having suffered paralysis