lmao no he's fine. there's medicine in the store if it's upgraded enough. they will get better on their own but they will give you a thank you gift if you consecutively give them medicine until they're better, like 2-3 days
Villagers will never die. Even if you don't give them medicine, they will still be alive. Not too sure on this, but the next day, they will probably still be sick, so you have another chance to give them medicine, so don't worry.
..Nah, just kidding. 2 of my villagers were sick at one point, and was cured after 3 days of medicine.
But if you dont give them medicine, they wont die. Your friendship with that villager will just lower.