He was born in Beaufort, South Carolina on August 10, 1801 into a prosperous and influential family. His father Robert Barnwell had served in the Continental Congress and the U.S. Congress. This Barnwell began his advanced education at Beaufort College, then graduated from Harvard. He returned home to manage the family plantation.

Robert Woodward's political career began in 1826 when he served in the South Carolina state House of Representatives for Beaufort County. He held that office until 1828, when he was elected to the U.S. Congress. He served as a congressman from 1829 until 1833. (He declined to run again in 1832.) From 1833 to 1841 he was head of the South Carolina College, now known as the University of South Carolina, in Columbia.

Barnwell was appointed to the United States Senate in after the death of Franklin Elmore on May 29, 1850. He served only from June until December, when after a special election Robert Barnwell Rhett replaced him. During this time period the tenuous balance between the northern and southern Senators required such short term appointments. His one distinction in the Senate involved the admission of California as a state. He opposed statehood in vain, but then had the good grace to introduce and present the credentials for one of her new senators, John C. Fremont.

In 1861 Barnwell was a delegate to the Confederate States of America Provisional Congress held in Montgomery, Alabama. At the congress' first meeting on February 4, 1861, William P. Chilton moved that Barnwell be appointed to preside temporarily over the Congress until its permanent organization. The Congress approved that proposal, but later that day, Barnwell handed the presidency over to Howell Cobb, who was elected president. In that Congress, he cast the vote (February 9, 1861) that ensured the election of Jefferson Davis as the Confederate President, and signed the Confederate Constitution. He represented South Carolina in the Confederate Senate from 1861 until 1865.

After the Civil War, he returned to Columbia and the University as an instructor. He as the chairman of the faculty at the South Carolina College from 1866 until 1873 when he retired. He died in Columbia on November 5, 1882 but was buried in St. Helena's Churchyard back in Beaufort.