Las Am?ricas International Airport
Air Antilles Express: Fort-de-France, Pointe-?-Pitre
Air Cara?bes: Fort-de-France, Pointe-?-Pitre, Port-Au-Prince
Air France: Fort-de-France, Pointe-?-Pitre
American Eagle: San Juan
Cubana: Havana
Dutch Antilles Express: Cura?ao, St. Maarten
Insel Air: Cura?ao, St. Maarten
JetBlue Airways: San Juan
LIAT: Antigua
Punta Cana International Airport
American Eagle operated by Executive Airlines: San Juan
Insel Air: Seasonal: San Juan
Miami Air: Seasonal: San Juan
Servicios A?reos Profesionales: Antigua, Aruba, Bridgetown, Cura?ao, Holguin, Pointe-?-Pitre, Port of Spain, St. Maarten, Santo Domingo-Las Americas, Varadero
USA Jet Airlines: Seasonal: San Juan
Cibao International Airport
Air Turks and Caicos: Providenciales, Seasonal: Grand Turk
American Eagle: San Juan
M&N Aviation: Seasonal: San Juan
Gregorio Luper?n International Airport
Air Century: Varadero
Air Turks and Caicos: Providenciales
Servicios A?reos Profesionales: Barbados, Beef Island, Holguin, Havana, Nassau
La Romana International Airport
Servicios A?reos Profesionales: Varadero
Saman? El Catey International Airport
Servicios A?reos Profesionales: Porlamar
Mar?a Montez International Airport
American Eagle: San Juan
La Isabela International Airport
Aerol?neas Mas: Aruba
As with the other postings related to your question on European destinations, this list too assembled from information Wikipedia, and thus may not be accurate.
And, yes, I included all islands in the Caribbean, regardless of their independence status, or their country status, because it is not clear to me what you consider “independent” and “country”.