What advice do you have for an in-law attacking my marriage? Could he ruin my reputation enough to get me kicked out?

It does not look like they can touch you. Allegations should be investigated and proven. My advice is to dump her and her family if they give you this kind of grief. The father is a control freak and is willing to play dirty pool. I see nothing but destruction if you have to deal with in-laws like that.

Your father-in-law says that you have caused stress to your wife and her family. What does your wife say about this? What is the rest of the story?

1) Be the best soldier, you can be. A good reputation can take a lot of heat, especially if it BS heat.
2) We don't know what "stress" your father-in-law is claiming you are causing, but unless it is criminal in nature or a violation of the UCMJ, it means nothing. So make sure you aren't involved in any criminal acts, or something that violates the UCMJ. Don't pick fights with him and don't let him get under your skin. His status as a Navy Senior doesn't carry a whole lot of weight unless you are doing something illegal.
3) It doesn't matter whether you are religious or not, chaplains have a lot of non-religious tools for dealing with extended family generated nonsense.
4) If you can't make her happy, don't try having a child as a solution.
5) Last but not least, ask your wife if she is unhappy. Make sure you get an honest answer. Then as best you can, do what it takes to make her happy. If it enough great. If it isn't enough, best to find out early.