Meriwether was born near Washington, Georgia, in 1806 and graduated from the University of Georgia (UGA) in Athens with a Bachelor of Arts (A.B.) degree in 1826. After studying law and gaining admittance to the state bar, he practiced law in Eatonton, Georgia. From 1811 to 1813, he was judge of the inferior court of Clarke County, Georgia. He was also captain of a volunteer unit during the Seminole Wars.
From 1831 to 1836 and again in 1838, Meriwether served in the Georgia House of Representatives as a representative of Clarke County. From 1845 to 1849, he served as a judge of the superior court for the Eatonton (Ocmulgee) district . In 1840, he was elected as a Whig Representative from Georgia to the 27th United States Congress and served one term from March 4, 1841 until March 3, 1843. He returned to the Georgia House in 1843, 1851 and 1852. Meriwether died on April 18, 1852, in Eatonton and was buried in that city's Union Cemetery.