Shippen was the brother of the Tory politician William Shippen. He was educated at Stockport Grammar School and Merton College, Oxford. He matriculated at Merton College on 6 April 1693 and graduated with a Bachelor of Arts degree in 1696.
Shippen acted as a tutor at Brasenose College, Oxford was awarded a Master of Arts degree on 4 July 1699. He was then elected a Fellow of Brasenose College. He was elected Professor of Music at Gresham College in London on 4 December 1705 and a Fellow of the Royal Society in 1706. He benefited from the living of St Stephen's, Limehouse. Shippen was elected Principal (head) of Brasenose College, Oxford in 1710 and attained a Doctor of Divinity. He held the post of Principal of Brasenose until his death in 1745. During his time as President of Brasenose College, Shippen was also Vice-Chancellor of Oxford University from 1718 until 1723.
In 1710, the same year he became Principal of Brasenose, he married Frances (died 1728), daughter of Richard Legh of Lyme Park, and widow of Sir Gilbert Clerke of Chilcote. Following this, on 3 October 1710 he resigned his professorship at Gresham College in favour of his elder brother, Edward Shippen (1671–1724), who had graduated from Brasenose with an M.A. in 1693, and M.D. in 1699.
Robert Shippen was buried in the chapel at Brasenose College, where he is commemorated with an epitaph (by Dr Frewin) and a bust.
- "William Shippen". Dictionary of National Biography. Wikisource. 1904. Retrieved 18 July 2011.
- Salter, H. E.; Lobel, Mary D., eds. (1954). "Brasenose College". A History of the County of Oxford: Volume 3: The University of Oxford. Victoria County History. pp. 207–219. Retrieved 18 July 2011.
- "Principals of Brasenose". The Brazen Nose, Volume 36. Brasenose College, Oxford, UK. 2002. Archived from the original on 9 June 2011. Retrieved 18 July 2011.
- "Previous Vice-Chancellors". University of Oxford, UK. Retrieved 18 July 2011.
- "Vice-Chancellors from the year 1660". The Oxford University Calendar. University of Oxford. 1817. pp. 27–28. Retrieved 18 July 2011.