It will likely not (at least not without running up roaming charges that would make Obama blush). Search cell phones, wifi, internet on here, there have been a lot of threads discussing these topics.

Generally, your best bet is to get Lime or Digicel service while you are down. If your phone is unlocked and will work on their networks, you can just get a sim card. Otherwise you can buy a cheap phone or there are apparently rental options available as well. From what I've seen, it appears that buying costs about the same as renting and you have the device for your next trip.