I've tried to source this claim by the always-interesting X account Traces of Texas. The histiography is a bit sketchy. But it's not unreasonable for North Street in Nacogdoches, Texas to claim the title of the oldest public road in the United States.
The Spanish explorers and early settlers referred to the route as La Calle Real del Norte. It stretched across eastern texas, serving as a trade out between a variety of indegenous peoples, most notably the Nacogdoche and Nansoni.
A historical marker in Nacogdoches, one of the oldest settlements in Texas, notes the possible beginning of the route.
Image: Google Maps