Coconuts is ok, but I think we can find you something really special. (I think it's a better fun-lunch-spot.) I know that The Reefs (and probably Cambridge Beaches and Pompano) will set up dinner ON the beach - candlelit table, toes in the sand...
Am assuming you're looking for sunsets and ocean views, so your best bets are the upscale restaurants at the hotels along South Shore - Elbow Beach, Fairmont Southampton, The Reefs, Cambridge Beaches and Pompano. You're going to get more privacy at the last three.
If you're looking for an in-town option, I really love Ascots at the Royal Palms Hotel - super service and a nice place to spend three hours enjoying dinner.

What a marvelous birthday present for your wife! You are sure to fall in love with beautiful Bermuda. There are a number of threads here which list activities and points of attraction, you may want to check them out. Where will you be staying in Bermuda?
For a romantic dinner, I would have to join txgirlinbda suggestion of the "on the beach" dinner done by Cambridge Beaches or Reefs. Very nicely done. If romance and privacy are your goals, I would call Cambridge Beaches and give them a free hand.

Agree with the above suggestions. Food at Cambridge beaches was better than Reefs when we visited a couple years ago.

We will be staying at the Fairmont Southampton.

Thanks for the great advice. I will definetly check into dinner on the beach at Cambridge Beaches.

Does anyone have any thought on the whole dinner on the beach idea? It sounds super romantic, but if it is windy, or cold I could see it not being as enjoyable...

The Reefs would be fairly close to your hotel, Cambridge Beaches is farther west.
I guess it all depends on how big an issue romance is in the equation. I would put a premium on great food (Cambridge Beaches, Ascots, Fourways, Tucker's Point) and a fairly elegant and quiet atmosphere. For my taste it is more fun to walk on the beach than eat there. But if you are looking for romance and something quite unusual, on the beach may fit the bill.

On other issue to consider is how your wife might choose to dress for a special occasion. Is it "beach" ready? Great slingback heels or a beautiful silk dress might not do all that well on the beach.

Out of the non-beach options, which has the best atmosphere? I would like something with a nice view, good lighting, quiet, spacious and not crowded...

Tough question:
Best view: Cambridge Beaches, Beau Rivage Tucker's Point.
Best food: Ascots Fourways Beau Rivage
Quietest: Fourways, pianist, Cambridge Beaches, fairly quiet in my experience, Ascots, very lively, Tucker's Point, depends completely on the crowd..there seems to be a good deal of "convention" business there. Beau Rivage is very busy and lively.
For a lovely view and very quiet (in my experience) option consider Splendido at Horizons.

At Splendido, you are on one of the highest point on the island and overlook the water near Elbow beach. I must have taken 25 pictures of the clouds over the ocean. We loved it there. We went on a recommendation or never would have known it was there.
Request a table outside..

Is Splendido still a great choice as I've heard that the Horizons hotel has closed? Has this changed the dining expereince at Splendido?

They are separately owned and managed. If it remains open after Horizons closes, I expect the quality to hold. I have not had dinner there, but had a lobster club for lunch which was absolutely terrific.

The most romantic place on the island, IMO, is the Fourways Inn. Impeccable food and service, elegant atmosphere, tables far enough apart that you feel like you can have a romantic conversation - we loved it.