A saying comes to mind here: "People may not always believe what you say, but they WILL always believe what you DO."
I'm not sure that you can say anything that will convince him that his suspicions are wrong, although it can't hurt to say that you have no intention of leaving him. I think the only thing that will convince him that you're going to stick around is....sticking around!
I'm curious, though...with as successful as you've been with low carb, has he ever considered joining you in your WOE and losing weight himself? He'd probably feel a lot more secure and he'd certainly be a doing a lot to ensure that you both have a lot more wonderful years together.
My DH and I have been married for 18 years as well. After I had been on low carb for 9 months, he decided to join me and work on shedding the 50 some pounds he'd gained in the [then] 16 years we'd been together. I often joke with him about not having to worry about divorce...It's taken me this long to train him and I wouldn't want to have to start over with someone else at this point!