and it's made me doubt myself and just feel so low. Maybe I did push him into cheating. Maybe I'm not capable of love
Typical gas-lighting. If you can't trace the path of the arguments origin, it may be what was happening here. If it wasn't, you were obviously wound very tight because------- HE BETRAYED YOU AND NOW WILL NOT GIVE YOU THE SPACE YOU REQUESTED.
This kind of abuse can lead to this kind of an over-the-top reaction. Please forgive yourself. I know a woman that went through the same as you, who is the most peaceful kind person I know, and she lost it when her cheater, living at home pushed her to that breaking point. You are not alone. Now that it has happened you seem insightful enough to avoid it again. In the end we control ourselves.
I recall going through the same stage as you, wondering if I drove him away. One of the reasons I didn't let him return when he asked was because I feared my own outburst and maybe my son would hear if he did. I am not an 'outburst' type person. Cheating partners can bring that out in a person. In your case, him refusing to leave yet, refusing to divorce, is vicious and cruel, not the actions of a loving person but the actions of someone worth divorcing asap.
There is a reason people here repeat over and over "cheating is not the BS's fault". Never. The reason this reminder is constantly needed, is the cheater 99% of the time twists realty to blame the BS, like yours just did.
He could have told you "Hey, dfdxb, you are a psychopathic, non-empathetic b**tch, either change or I want a divorce" back when you had one of those WAY BACK IN ALTERED TIME earlier arguments. If what he says is true, why would he stay with you, lie, mess with your head, and bang someone else. It makes no sense. He would have been wanted a divorce to get away from you. At the very least, even if you had your own issues to work on (all people do), he is a shitty husband and a divorce would make sense even on that level.
Be patient with yourself, these circumstances suck. No contact going forward is the only way to go, especially in the same home. Yes or no answers if needed. No emotion. He does not deserve to speak with you. And one more time, you didn't make him cheat. That's like saying you made someone steal or murder someone. It doesn't work that way.
It is worth looking at your doubts for your own self-growth, but his choice to deceive you proves he is not worthy of joining you any further on your path. Take care.