Well, you'd need a passport if there was enough time before the continuation AND the host country decided to let you go through immigration/customs.

Normally however, you cannot be held responsible for a lack of travel documents for a diverted flight. Regional immigration/customs supervisors normally have the authority to make modifications to documentary requirement in urgent cases. For example, if a full 777 from NYC to Anchorage was diverted to Yellowknife up far northern Canada with no replacement transport possible for 4-5 days, the tiny local airport obviously could not host that number of people for that long and they would have to be sent to local hotels. Passengers would be required to provide the photo ID they boarded with, but they would be allowed out. This would require senior level immigration manager sign-off, but it could and would be done in an urgent situation. Hope that helps.