You may not be able to hang out your shingle but your services won't be refused. Most of the islands in SoPac, once you get away from the capitol, have no full time medical
staff. The services of a traveling physician will be in demand. You may get paid in coconuts, but you will receive the gratitude of the locals and be adopted by everyone. You would have to bring your own pharmacy and diagnostic tools will be a stethoscope. In short, medicine as practiced in the 19th century.
There was a DO cruising when we were in French Polynesia
. He wasn't looking for work but managed to stay as busy as he wanted and they never had to worry about their next meal or even preparing it.
Unsolicited, another cruising doctor stopped me on the Quai in Papeete and remarked about a small staph infection on my knee. I'd been fighting it with successive rounds of Tetracycline for the previous six months but it kept coming back. He said to treat it with a weak bleach solution which I did. The infection was gone in a couple of days never to return. Earned my gratitude.