I think a few days in New York should suffice for visiting the must-see places. After that you should travel to at least one other part of the USA, if only to get a more balanced picture of this big country. Besides, after the intense experiences of the city, you'll be probably look forward to the quiet of the less-populated places.
Two places I would recommend that fall within your criteria of no more than a day's drive away are the Appalachian Mountains (along with the Shenandoah Valley) and the coast of Maine.
Great Smoky Mountains National Park is a little further away (10 to 12 hours) and, while it offers some most impressive views, it's likely to be filled with other tourists this time of year. But, there is Shenandoah National Park and several national forests in Virginia that are nearly as great. And, by all means take in the Blue Ridge Parkway. Unless the weather is really terrible, you won't be disappointed. Even then, it's pretty awesome in a rainstorm.
If you choose to venture north, I agree with so many others in recommending Acadia National Park. Like in the Smoky Mountains, it probably will be crowded, so figure to be there mid-week. Since Fourth of July falls on a weekend, you'll probably be okay. You should enjoy Acadia for a day or two, but I think you will have even more fun exploring the small towns of the Maine Coast. Plan to take US 1 at least one way. The stretch from Freeport (just north of Portland) to Ellsworth (close to Bar Harbor) offer countless possibilities. If you have the time, take some side trips down the peninsulas for fishing villages, old farms and rocky coastlines.
I hope you get to at least one of these destinations, even better if you can enjoy the two.