“in hospital” or “in the hospital”?

‘He’s in hospital’ and ‘He’s in the hospital’ both mean that he is in a hospital for examination and/or treatment.

‘He was hit by a car and now he’s in hospital’ would tend to suggest that he is in there for some time while he’s recovering (for however long that takes).
‘He was hit by a car and now he’s in the hospital’ could mean the same, but is more likely to be used when he has been taken to hospital for a check-up, or possibly he will be there overnight and be released the next day.

‘in hospital’ is longer term than ‘in the hospital’ which is either known to be temporary or, at least, not yet known to be long-term.

As Atomos pointed out, ‘He is in the hospital’ could also be used as a reply to ‘Where is the doctor?’ (in the same way as you could say ‘He’s in the house’) but a more likely is ‘He’s at work’ or ‘He’s at the hospital’.