While a healthy diet can't hurt, there are zero diets (including the famous anti-cancer diet) that clinical trials show prevent cancer from coming back. There are studies showing that there are very bad diets that can encourage cancer like lots of red meat and sugar. here is a link to the major studies on antioxidants from the National Cancer Institute
Antioxidant fact sheet
The reason to go on a healthy diet is to improve your vitality and your health and prevent obesity. It it makes you or your dad feel better to believe that a healthy diet will prevent cancer from coming back that's a bonus but not a guarantee.
Oh, I almost forgot, taking antioxidants or an antioxidant diet during treatment HELPS the cancer tumor so don't start until you get the all clear from your doctor
Charm