He was born in London on 1 January 1676; his father, Charles Gravener, a prosperous upholsterer, lived at the Black Swan on Watling Street.
At the age of 14, he was baptized by Benjamin Keach and subsequently admitted to his Particular Baptist congregation in Goat Yard Passage, Horselydown.
In 1699, Grosvenor underwent examination and licensing by seven Presbyterian ministers, including Robert Fleming the younger, and became assistant to Joshua Oldfield at Globe Alley in Maid Lane, Southwark.
Gravener's reputation as a preacher grew, and following the death of Samuel Slater on 24 May 1704, he was chosen as pastor of the Presbyterian congregation in Crosby Square.
Throughout the years, his assistants included Samuel Wright (1705–1708), John Barker (1708–14), Clerk Oldisworth (1715–26), and lastly Edmund Calamy IV (1726–49).
At Salters' Hall he lectured against popery in 1735, taking persecution as his theme; and he was active in the Old Whig, run 1735–8 by Benjamin Avery.
This serial continued till 1719, and was influential on the subject of religious liberty, and with the non-subscribing majority at Salters' Hall in 1719.
Grosvenor is said to have drawn up the Authentick Account (1719) of the Salters' Hall proceedings, the first of the many pamphlets issued by the non-subscribing ministers, with a list of names.