No, the nearest you will get is to drive east until you meet the north end of the Saman? highway between Nagua and S?nchez. Then drive south along that highway to Las Americas, and head east from there to Punta Cana.
In theory you should be able to drive all the way to Saman? and take the ferry to Sabana de la Mar, and from there drive on southeastwards via El Seibo or Miches, but you would have to leave your car behind in Saman? because the ferry is for foot passengers only.
