When he "doesn't believe in therapy"

Let's start with this: Therapy isn't something one "believes" in, like Bigfoot or fairies or alien abductions. We have data that therapy exists. So when someone says they don't "believe" in therapy, what he's really saying is he doesn't believe that talking with an objective person about his life and his problems is going to help him.
To which I say, "really? Tell me more about how therapy won't help you."
Therapy is, of course, a broad term for a whole lot of approaches to helping people move through problems that are getting in the way of leading a productive, healthy life. Not "believing" in therapy is a cop out. Far more truthful to say, "I don't want to go."
To admit that he doesn't want to go, however, opens your husband up to your disappointment or your anger and your frustration. He'd prefer to hide behind the fiction that therapy won't work for him so why not save time and money by not bothering with it at all. After all, he'll tell you, it's bogus. Or he doesn't like to talk. He'll refer to therapy as "woo-woo" head shrinking stuff for crazy people. Surely not for someone as sane and feet-on-the-ground as a guy like him whose only problem is that he violated his marriage vows, lied to his partner and risked losing a marriage that he now claims he never wanted to lose.
That avoidance of discomfort is exactly the kind of behaviour that got him into this situation. By not being forthright and honest, he created this shitstorm that he now wishes would just go away without him having to do anything that he doesn't want to do. Or rather, that he doesn't "believe" in.
I came to therapy reluctantly. I grew up in what my therapist calls a "distressed home" with plenty of addiction, depression, anxiety, suicide attempts and strife. When my mother got sober, I was 20 and launching into my life. She wanted me to go into therapy because she, correctly, saw that I held a lot of anger about my earlier years. I refused, insisting that I was "fine" and that I didn't need some stranger telling me how to feel.
A few years later, however, as I struggled within a highly toxic relationship with a man I couldn't imagine living without, I relented.
I thought she could help me figure out how to make this guy love me enough to stay. Instead, she helped me find the self-respect and strength to leave.
Which could be a big piece of why your husband doesn't "believe" in therapy. There's huge fear for many people in discovering just what's lurking in their own hearts, and that of their partners. For those people, pretending that everything is fine is preferable to knowing it isn't. They might even have convinced themselves that, all evidence aside, everything is fine. If only you could just stop talking about their mistakes, about the destruction they've caused. After all, they won't do it again.
Until they do.
Until they come up against something in their lives that they simply don't have the tool in their emotional toolbox to handle. All of us lack certain tools. I don't know a single human being whose parents were able to provide them exactly the number and type of tools they would need to handle whatever life throws their way. Some of us can develop our own healthy tools, but far more of us either rely on crappy rusty tools to cope – we drink, we shop, we ignore, we rage, we cheat – or we fall apart completely.
Somewhere in there, the smart ones among us say one thing: Help.
We realize that if we were so awesome at solving our own problems, we wouldn't be in this mess. We acknowledge that our way of coping has created some highly unpleasant side-effects, like a wife whose eyes hold a world of pain that we caused.
And then, the smart and courageous ones allow themselves to consider that maybe, just maybe, this therapy thing is worth a try.
It might not work with the first therapist. It might require a few tries. But my guess is that these same guys would continue to find a good mechanic for their car if the first one didn't seem to great rather than decide that they don't "believe" in mechanics.
It will undoubtedly require a lot of ego-checking and patience as everyone finds their footing and begins to establish an atmosphere of trust. After all, you should all be there for the same reason. To create a healthy relationship based on honesty and respect and compassion. 
Because that, whether or not these guys will admit it, is what everyone is after. And, too bad for them, part of that process is going to require that this guy who doesn't "believe" in therapists, has to dig deep into his psyche and figure out why he risked everything that mattered for something that didn't. Or at least didn't matter as much.
He should want this. He should be willing to do whatever it takes to begin to heal this damage he created. He should be willing to make himself uncomfortable in order to help you feel safe again.
If he won't? If he continues to hide behind this fiction that therapy requires "belief" rather than hard work, then he's telling you that this marriage isn't worth the effort required of him.
This is painful but crucial information for you to have. Because it makes your choice – whether to stay in the marriage with full awareness of how much effort he's willing to put into rebuilding it, or whether to leave with that same awareness – a lot more clear.
I'm not insisting that no marriage can be saved without therapy. I am saying that I don't know of any. Sure, I know of marriages that survived infidelity without therapy. But I don't know of solid happy marriages that have. The solid happy marriages I know of that have survived infidelity have done so with a team of support, from friends to, yes, therapists. Personal therapists, marriage therapists, family therapists. Cognitive behavioural therapy, EMDR, couples counselling.
So while it's possible that a marriage can be rebuilt without the help of counsellors to guide couples toward healing, I don't "believe" it's helpful to anyone to ignore the valuable assistance of an objective, experienced therapist.