The fair place to meet is Kansas City, MO — the city closest to the midpoint of Denver and Raleigh. From the farther side that’s about 18 hr 56 min of driving.
Recommended midpoint
Kansas City, MO
From Denver
11 hr 41 min
557 mi to Kansas City
From Raleigh
18 hr 56 min
903 mi to Kansas City
Denver has the shorter trip; the split is off by about 7 hr 15 min. The alternatives below can even it out. Denver and Raleigh are about 1,460 miles apart.
Denver and Raleigh are about 1,460 miles apart by road. Split the difference and you arrive near Kansas City, the city closest to the halfway point between them. That puts roughly 11 hr 41 min of driving on the Denver side and 18 hr 56 min on the Raleigh side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Kansas City still makes a fair, central place for Denver and Raleigh to converge, splitting the travel instead of asking one side to cross the country.
If Kansas City doesn't have what you're after, Memphis and Nashville are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Denver and Raleigh.
Kansas City, abbreviated KC or KCMO, is the largest city in the U.S. state of Missouri by both population and area. It is located on the Missouri River at its confluence with the Kansas River, within Jackson, Clay, Platte and Cass counties. Read more →
Denver is the capital and most populous city of the U.S. state of Colorado. Officially a consolidated city and county, it is located in the South Platte River valley on the western edge of the High Plains, and is just east of the Front Range of the Rocky… Read more →
Raleigh is the capital city of the U.S. state of North Carolina. It is the second-most populous city in the state, tenth most populous city in the Southeast, the largest city in the Research Triangle area, and the 39th-most populous city in the U.S. Read more →
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.
Estimates use straight-line distance and typical road speeds; real drive times vary with route and traffic.