Keep feeding whole. The way to do this is to only have a whole grain bread option. He will eat it, or not eat bread at all. You simply stop buying anything else. Do not worry at all about his lack of bread in his diet! Eventually over time he will eat it.
Persist and conquer. Just keep offering and be a good example. Many times it takes a ton of times offering before a child takes what you wish them to eat. Some of that is age and relates to taste and textures and some is just built in preference and pickiness. But most kids will eventually try foods presented many times. Try asking a child to take one bite of that food each time and go from there.
Keep offering. Many times it takes introducing a food 10-20 times before a child will take it. In the meantime don't begin offering white bread- this shows them there is an alternative. Instead, continue offering whole grains, but don't force children to eat them- you don't want it to seem like a punishment. Offer other foods as well- veggies, fruits, etc. Pretty soon your child may try it and may like it!