Hi Charles!  Walt Disney World is not just for kids!  Once you're there without the kids, you'll naturally look at everything from a couple's perspective instead of from a child's perspective.   Just holding hands on an attraction without your children around you can be romantic.
 You might consider the quiet elegance of 
Victoria & Albert's located in Disney's beautiful Grand Floridian Resort & Spa.  Check out additional information on 
Restaurants at Walt Disney World Resort here.  Some of my other favorites for romance are California Grill at Disney's Contemporary Resort, Jiko - The Cooking Place at Disney's Animal Kingdom Lodge, and Monsieur Paul in the France Pavilion in Epcot Theme Park.  As a matter of fact, strolling leisurely through the countries in Epcot at night is a wonderful thing to do as a couple.  
For even more magic, why not book a romantic Couples Massage at 
Senses Spa at Disney's Grand Floridian Resort.  Afterwards,  stroll the grounds of the Resort and grab a hammock near the beach.  
Be sure to ask the folks at Theme Park Guest Relations for a set of "Just Married" buttons.  Cast Members are great for providing unexpected surprises when they see these, even if it's just a special "shout out."  
Don't forget to watch Epcot's IllumiNations: Reflections of Earth and Magic Kingdom Wishes Nighttime Spectacular together.  You might even like to book a 
Specialty Cruise for that extra something special.
 I wish you both a MAGICAL honeymoon!
-Jacqui