If you want to go to a non-commercial private island, your best bet in the US is Cumberland Island. The Greyfield Inn is the only hotel and, in fact, the only business on the entire island. At the Carnegie owned mansion turned into a charming Inn, some of the rooms have shared bathrooms. Breakfast, picnic lunch and dinner are included in your package. Also, you can get a private guided tour of the inn.
It is rated among the Top 10 Romantic Hotels in US, and was the site of the John F. Kennedy, Jr.- Carolyn Bessette wedding in September 1996.
While on Cumberland Island, you can enjoy very private beaches where you will see feral horses running around. It is also nice for bird watching, fishing, biking and kayaking. See the turtles hatch in the summer.