You say "counseling is out of the question" for HIM.
But not for YOU.
Please, please get yourself to a counselor or therapist as quickly as possible, even in the next few days if you can. This is a very fragile marriage in a potentially devastating situation. You seem to be handling it pretty well, but that is only "handling it" for the short term, and not the same as resolving and communicating better in the long term. You need to figure out your feelings; what you want him to do or not do next; whether you want to insist that this woman get a paternity test so you KNOW if this is his child (what if the baby isn't his after all? That settles one thing but not the issue of his sleeping with her, about which you may have more feelings than you now realize)...There seem to be some issues here too, based on the rest of your post, about the fact he's from this male-dominated culture and has a distant-to-nonexistent communcation style.
Ask yourself: Even if this one night stand and the child were NOT in the mix at all, would you still say you are really completely happy with him, your marriage or your communications? If the answer is no -- and I suspect that's the answer based on what you describe as a rather up-and-down abilty to really communicate -- then you should get help from a professional.
Get counseling, intensive if need be, for yourself, and maybe that counselor can help you figure out how to get him into couples counseling...Because frankly unless he learns to communicate better with you, I would think there is a chance of further, serious problems beyond even this gigantic bombshell. If he feels depressed/distant again, he may once again turn to someone else. I say that not to scare you but to point out the reality that people who cannot communicate with others, who are inwardly focused, and who have the added burden of depression may be more likely to do things that are risky. If he can't change these things, or at least sincerely work on them for your sake and your child's sake, then he runs the risk of further risky or impulsive behavior. I've seen depressed spouses do selfish and risky things that hurt or even broke up their families -- you have to find out if he is willing to stop himself by getting help.
(By the way, if he is depressed, do you mean he's been medically diagnosed as clinically depressed? How can that be if he won't see any form of therapist? If he just says he feels depressed but has never seen a doctor to get a diagnosis, then he must see one. And the doctor should not just medicate him but should send him for talk therapy as well. One suggestion: Would he be more likely to talk to a doctor from his own cultural background, and one who is male? It sounds like he might have the cultural attitude that "Men do not share problems, men are always strong and silent, and men do not admit to being weak by sharing weakness with others, especially women." A LOT of men of many cultures have this unspoken issue with seeking therapy or other emotional or mental help. So it might help some if you could find a doctor who "gets" that.)
Please, get to a counselor for yourself as soon as you can and please update us here. I feel awful for you and I know you don't want to hear "Counseling" but I don't see how this complex situation, with a spouse who is not communicative, can be worked on otherwise. Take good care of yourself in the meantime, and stay as healthy as you can, for your child.