He was then apprenticed to John Fothergill, a medical practitioner at Darlington.
He also made a large contribution to the understanding of aortic dissection by publishing several case series on the condition. At the age of nine he was sent to the boarding-school of Samuel Marshall at Kendal. He then travelled for his health, twice visiting Ceylon, and studying for a time at Paris.
He spent 1838 as house-surgeon to the hospital at Chester, and in 1841 went to the University of Edinburgh Medical School, where in 1842 he took the degree of Doctor of Medicine In 1844 Peacock was admitted a licentiate of the Royal College of Physicians of London, and in 1849 was elected assistant physician to Saint Thomas"s Hospital.
In 1850 he was elected a fellow of the College of Physicians, and in 1865 delivered the Croonian lectures there on Some of the Causes and Effects of Valvular Disease of the Heart. A dispensary which he began in Liverpool Street, London, ultimately grew into the Victoria Park Hospital for diseases of the chest, to which he was physician from its foundation.
He lectured at Saint Thomas"s Hospital, and was one of the founders of the Pathological Society of London in 1846, and a contributor to its Transactions. He was its secretary in 1850, vice-president 1852-1856, and president in 1865 and 1866.
The College of Surgeons gave Peacock a gold medal in recognition of his valuable additions to their museum.
He lived at 20 Finsbury Circus, London, an area popular with physicians at the period. He had an attack of left hemiplegia in 1877, but went back to work. In 1881 he had a slight attack of right hemiplegia, from which he also recovered.
On 30 May 1882, while walking in Saint Thomas"s Hospital, he became suddenly unconscious, fell in one of the corridors, was carried into a ward which was formerly under his own care, and died there the next morning, without having recovered consciousness.
In 1850 Peacock married Cornelia Walduck, also a Quaker, who died childless in 1869.
In 1833 Peacock went to London as a medical student at University College, also attending the surgical practice of Street George"s Hospital. And in 1835 he became a member of the College of Surgeons and a licentiate of the Society of Apothecaries.