Can I be treated in an emergency onboard?

Most cruise ship clinics are for minor incidents and temporary illnesses. They are not equipped for major surgeries or other pressing medical care needs. In case of serious illness or injury, you would need to be transported. That means either back to the US or to the nearest foreign facility, by air. In this case, you will need to have a passport.
 
If you neglected to take out travel insurance with medical coverage, be aware that most US medical insurance only covers you on US soil. Unlike our own military seafaring vessels, the majority of cruise ships are not “US soil”, even if the cruise line itself is a US operator; the ships are registered in places like the Bahamas, Singapore, Malta, etc. . So, again, serious treatment means being flown somewhere. All border crossings by air into any country require a passport. Period.
 
Do be sure to make copies of your passport to include in each piece of checked luggage, to leave with a family member at home, and to exchange with your travel partner. Take a photograph of the passport number with your cell phone. These measures help with identification of misplaced luggage and other inconveniences, but are no substitute for the original document in an emergency.