Just my two cents here, but I find myself more prone to complain about little things when something much bigger is bothering me, and I just can't seem to deal with it. It seems that your husband's biggest problem is being a chronic complainer, and I would expect that he also has something bothering him. It seems as though it's been bothering him for a long time, and he has completely despaired of ever resolving it.
One thing you can't do, however, is bring it up directly--even if you have a strong suspicion about what it is. This would be seen as a frontal assault, and would be met with the strongest defense possible. If you really want to deal with this it will take lots of patience, because you cannot introduce the subject with any indication that you suspect a problem, or that you feel he ought to open up to you on this, etc.
For example, you mention that he left the Catholic church when he married you, and he may feel guilty about this. Note that I'm not saying that he does, only that he may--you will have to use your intuition here and think of something that *might* be bothering him. Look for an opportunity to bring up the subject in a roundabout way--the more roundabout the better, like something in the news, or a local event that you heard about. This is where you'll get an honest reaction, and if it's particularly hostile, it's a good sign that you're "warm."
Then, just let it drop. If you act too interested, it will be obvious that you're "fishing," and he will just clam up. In fact, I'm not even sure what you would need to do next, except to pray over it and look for an opportunity to bring up the subject again.
(One thing you need to be aware of, too, is that he knows how you typically react in emotionally charged situations, and probably doesn't like it at all. This is a common reason men have for emotionally shutting their wives out. If you can notice any patterns in you own behavior and manage to alter them, you may find that he starts behaving differently.)
May God bless you both, and may He bring healing to your marriage.