The North American country home to the oldest continuously inhabited European settlement in the Americas is the Dominican Republic.
That settlement is Santo Domingo, founded by the Spanish in 1496 and associated with Bartholomew Columbus. An earlier Spanish settlement on Hispaniola, La Isabela, was founded in 1493 but abandoned around 1500, so it does not meet the “continuously inhabited” condition.
Santo Domingo became the first permanent European colonial center in the Americas and the first seat of Spanish colonial rule in the New World. The city later moved to the west bank of the Ozama River, where its Colonial Zone preserves buildings and streets from the early sixteenth century.
The claim is about a European-established settlement, not the oldest human settlement. Taíno communities existed on Hispaniola for centuries before Spanish arrival. Santo Domingo is also associated with several “firsts” in the Americas, including the first cathedral, university, monastery, and fortress built under European colonial rule.