Newnham was commissioned lieutenant on 6 December 1760, and from 1769-71 commanded the cutter Hunter 4 off the Isle of Wight, paying her off in July of the latter year. He was promoted commander on 24 March 1779, joining the sloop Star 10 in the Leeward Islands, and was posted captain on 30 July.
Having been immediately appointed to the Grafton 74, he was flag-captain to Commodore Thomas Collingwood at the Battle of Martinique on 17 April 1780, being wounded in the left thigh and left hand for which he was allowed a 50 pension. Two of his men were killed and another twenty-nine wounded during that engagement. He commanded the Grafton in the early stages of the ensuing Leeward Islands campaign, including the follow up engagements in May when his ship did not suffer any casualties, but he left her shortly afterwards
Newnham was appointed to the newly commissioned Africa 64 on 25 April 1781, serving in home waters from July until a December refit at Plymouth, whereupon he left the ship in the following January. He thereafter commanded the Defence 74 from 13 January 1782, departing for the East Indies during the following month with Commodore Sir Richard Bickerton s squadron, and in which he fought at the Battle of Cuddalore on 20 June 1783, losing seven men killed and thirty-eight wounded. He removed to the Isis 50 on 23 December and returned home to be paid off in July 1784.
In the autumn of 1793 he recommissioned the Saturn 74 but left her in the ensuing August.
Newnham probably died in early 1795. His address was given as Aldershot, Hampshire, in which town he was buried.
In June 1785 he married the widow of Admiral Clark Gayton, Elizabeth Legge, at Fareham, the admiral having died earlier in the year. She subsequently married the Reverend James Pigott in 1801.