Spaight was born in New Bern, North Carolina, the son of the Secretary of the Crown in the colony. Orphaned at the age of eight, Spaight was sent to school in Ireland and graduated from the University of Glasgow.

In 1778, Spaight returned to North Carolina and served as an aid to General Caswell during the American Revolutionary War until 1781. Although he was elected to the North Carolina General Assembly in 1779, he was denied his seat after the election was challenged. Spaight was again elected in 1781 and served until 1783.

The General Assembly elected Spaight a delegate to the Continental Congress between 1782 and 1785; he then served in the North Carolina House of Representatives from 1785 to 1787, and was named Speaker of the House.

On 24 March 1788, Spaight married Mary Leech; their son Richard Dobbs Spaight, Jr., later became Governor of North Carolina in 1835.

Under the new North Carolina Constitution of 1787, Spaight was nominated for Governor, but was defeated; he was nominated for the United States Senate in 1789 and was again defeated. In 1788, he was a member of the state convention which voted not to ratify the United States Constitution, although Spaight himself supported ratification.

Spaight retired from politics for several years due to ill health; he returned to the state House of Representatives in 1792, was elected governor that same year, and re-approved by the General Assembly for two further two-year terms.

During Spaight's term as governor, sites were chosen for the new state capital of Raleigh and the newly-chartered University of North Carolina. Spaight served as chair of the university's Board of Trustees during his term as governor. Spaight stepped down as governor in 1795, having served the constitutional limit of three one-year terms.

Spaight was named to the United States House of Representatives in 1798, filling the unexpired term of Nathan Bryan; he was elected to a two-year term in 1799, serving until 1801, and associated with the Democratic-Republican party of Thomas Jefferson. He declined to run for another term in Congress, but returned to state government, serving in the North Carolina Senate beginning in 1801.

Spaight died on 6 September 1802, following injuries sustained in a duel with John Stanly, the Federalist Congressman who succeeded him in the House of Representatives. Spaight is buried in his home town New Bern.

Spaight Street in central Madison, Wisconsin is named in honor of Richard Spaight. Most of the main streets in downtown Madison are named after signers of the United States Constitution.