Barbados or the west side of Puerto Rico has both and you can do it in the same day.
I second Barbados. I currently live here and when there is a good swell, there is great surfing on the east coast. Kelly Slater loves coming to Barbados. I would say the diving is good, but honestly have only been diving on a few of the other islands in the Caribbean, so don't have much to compare to. If you keep your eye out, you can get good deals on flights as well from NY.