Can hospitals discharge a patient in intractable pain?

If the pain does not require medical supervision, yes, hospitals can discharge the patient. When this happens the patient is normally prescribed medication, which they can take at home.

Hospitals do not have the space, resources and staff to watch people 24 hours a day, 7 days a week. So if a patient is deemed medically safe to leave the hospital on medication, they can be discharged. Discharges are not made on a whim, they are discussed professionally by doctors and the patient is only discharged if all the doctors in the discussion agree with the discharge.