We have gone to Hilton Head for July 4 almost every year since the Disney resort opened! It's one of our two favorite times there (the other being Thanksgiving).
The resort has always done a lot of great activities, beginning with bike decorating in the morning for a parade through the resort in the afternoon. There are also pool games at the beach house in the afternoon. The poolside BBQ, which is normally held on Tuesday (Star Spangled Tuesday), is on the July 4 (whatever day of the week that might be). They leave the pool open later in the evening, with poolside dancing and contests. It's good old-fashioned fun!!
You can see the fireworks right from the resort -- no need to go anywhere.
And right across the marina from the resort, Shelter Cove has a festival going on as well, that you can easily walk to.
I would recommend, however, that you not leave the resort later than mid-afternoon. It starts to get very crowded near there, with people from all over the island coming to the area for the fireworks. (There are guards making sure that no one gets into the Disney resort area without a parking pass.) If you're not doing the BBQ at the resort, this is probably NOT the best night to go out to dinner.

And if you're doing your own cooking for the evening (the resort has grills scattered throughout the grounds), get everything you need earlier in the day.
This may be the last 4th of July we get to participate in the resort activities, though. We are in the process of purchasing a villa very near the resort so that we can visit more often and bring our dogs with!
