What is the ruling on someone who made a specific vow, thinking he was afflicted with a disease, and then it became clear that he was not afflicted with it? And is he obligated to fulfill the vow if his disease is incurable?

Fulfilling a conditional vow is only obligatory if the condition it was tied to is met. So, if you were not sick or did not recover from your illness, then you are not required to fulfill it. This type of vow is disliked by most scholars, although it becomes obligatory to fulfill if its condition is met.