Mossman has a fully operational hospital with a 24 hour emergency department 

General hospital

https://www.cairns-hinterland.health.qld.gov.au/hospitals-and-health-centres/mossmanmulti-purposehealth-service

A variety of medical centres provide primary care and allied health services as well as visiting specialists.

Port Village and Mossman Medical Centres (Mossman & Port Douglas)

https://www.portdoctors.com.au/

New Horizons for Health (Port Douglas)

https://www.newhorizonspd.com.au/