I really am not sure about fasting as a diet plan, seems like a plan that tries to spiritualize starving yourself for weightloss.
Now to fasting, Number 1 its none of the diet programs business whether you fast or not, infact Christ says we shouldn't be telling people of our fasting in general.
I had the same issue with my wife actually, not really an issue sometimes we feel lead to do something that our spouses just don't understand, so what did I do to ease my wife's mind on my health? For one I told her I would always inform her of when I was fasting and for how long I was going to fast. I am hypogylcemic and so if I don't eat something with sugar I not only get grumpy but I potentially can pass out, solution? When I fast I usually drink juices and milk to keep me going.
If you husband is not only with day long fasts and beyond that maybe he would be more comfortable with you just fasting a meal for a selected day. By any means its important to start slow and work your way up.
I do question one thing, Some people can't fast? This wouldn't stand really in light of God calling the entire nation of israel not only to a one day fast but to a 40 day fast, in which everyone fasted.
Do some research on fasting and you will be surprised to find that it can benefit the body, for one your always much better off with one less cheeseburger in your system, this is one advantage is your body with a long enough fast will detox itself from all the junk we usually eat. I understand your husband concern because my wife had the same but if done properly fasting can be really great for you. Also if he is super concerned about it go see your doctor and find out if your body could handle fasting.
If your doctor says no, then I would say don't do it and find other things you can fast, such as internet, televion, the news, etc
Getting your family comfortable with the idea of you fasting is greatly dependant on you learning researching and doing it the right way before ever pursuing it this tends to ease worried minds.