The famous surgeon was born in North Shields. He studied at Edinburgh and had a distinguished career in medical life. In 1820, Greenhow married Elizabeth, elder sister of Harriet Martineau, the famous writer and granddaughter of Mr Rankin, the sugar refiner of the Close in Newcastle. The couple moved into 1 Eldon square in 1827. Together with Sir John Fife (q.v.) Greenhow established the Newcastle Hospital for Diseases of the Eye. He also proposed a university for Newcastle in 1831, and was an enthusiast for smallpox vaccination. In 1832 Greenhow became Surgeon, and later senior Surgeon at Newcastle Infirmary, where he remained for some 23 years.
Greenhow was of an inventive turn of mind and devised a fracture bed for supporting broken legs. This was used for many years at the infirmary on Forth Banks, and later at the Royal Victoria Infirmary. Greenhow became one of the first surgeons to become FRCS (in 1843). Greenhow and Harriet were on affectionate terms and she often stayed in Eldon Square. When brought low by her mysterious illness in 1839, it was to be near him that she took lodgings at 57 front street, Tynemouth, where she stayed for some five year. Harriet's recovery through mesmerism led to a breach and she moved to Ambleside for the last thirty years of her life. In her Life in the Sick Room, Harriet writes with regret:

'Her home at Newcastle [Elizabeth's] with all possible kindness from her hospitable husband and herself, was always at my command, without hindrance or difficulty until my recovery from a hopeless illness in 1844 by Mesmerism proved too much for the natural prejudices of a surgeon and a surgeon's wife, and caused by the help, or the ill offices of another relation, a family breach as absurd as it was lamentable.'