Praise be to Allah.
Chewing gum is not free of substances that dissolve when chewed and enter the stomach, so it is not permissible to use it during the day in Ramadaan when one is fasting. He could do without it and do exercises for the jaw instead, and chew gum after the sun sets and until dawn comes.
If he can find a kind of gum that does not contain substances that dissolve when it is chewed, then it is permissible to use that because it will not break his fast, because nothing from it will enter the stomach. But your father should be advised not to chew it in front of people who do not know about his situation and his excuse, so that they will not accuse him of neglecting his religious commitment. If this kind of gum cannot be found or your father needs to chew the regular kind of gum during the day, and if not doing that will lead to a delay in his recovery or to his sickness becoming worse, then it is permissible for him to break his fast in Ramadaan, and he should make up the days that he misses, because Allaah says (interpretation of the meaning):
“and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days”
And Allaah knows best.