Born in Providence, Rhode Island, Pitman was educated at Brown University, receiving an Bachelor of Arts in 1799, at the age of fifteen.
He read law to enter the New Hampshire Bar in 1805, and the New York Bar in 1806. From 1806 to 1821, he engaged in the private practice of law in various locations, including New York City (1806–1807), Kentucky (1807–1808), Providence, Rhode Island, (1808–1812 and 1820–1821), Salem, Massachusetts (1812–1816), and Portsmouth, New Hampshire (1816–1820). From 1821 to 1824, he was the United States Attorney for the District of Rhode Island.
On August 4, 1824, Pitman received a recess appointment from President James Monroe to a seat on the United States District Court for the District of Rhode Island vacated by David Howell.
Pitman was formally nominated on December 16, 1824, and was confirmed by the United States Senate, and received his commission, on January 3, 1825. Pitman continued in that office for 40 years, until his death, in Providence, Rhode Island.