I'd like to point out the good and the bad (on your side) here. [The bad on your wife's side is so obvious I'm not going to mention it!]
DH: "What are you trying to prove here?"
That is actually a decent question. It is a bit provocative, but it is also a bit validating--you are at least showing genuine interest in her, her thoughts, and her feelings.
DHw: "I'm trying to prove that you're despicable"
Ugh. That is hard to hear. But it was a baby step in the right direction--she was telling you something about her real feelings. It was a tiny bit vulnerable and honest with you... .and also is waaaaaay more open than a month of not letting you even touch her.
This is a really hard statement to validate, but if you succeed, it could thaw a lot of ice. Think about she is feeling when she says that, and remember that her FEELINGS are valid. Even if her actions aren't valid.
That she feels you are despicable is a valid feeling. Or perhaps that she feels you hurt her, or is afraid you will say something to hurt her which might be closer to the target is valid. The conclusions she jumps to aren't valid--that she has to 'prove' it by treating you badly, or that you 'deserve' the punishment.
So for the future if she says something like that, think about what the valid feelings behind it are and whether you can validate them. Or perhaps you can ask her more about them, which is also validating. Here are a few things to consider:
"What do you feel I've done that is despicable?"
"You sound angry with me... ."
"You sound really upset with me."
And remember--sincerity matters. If you are feeling hurt and defensive and falsely accused, that will come through non-verbally and it won't be validating.
Only ask these questions if you can find a sincere curiosity about what she is feeling that you are asking about--and are ready to keep directing the conversation back to her (genuine, valid) feelings, and away from her (abusive, likely imagined/fabricated, or projected) judgments and conclusions about you.
DH (Shocked that she would actually admit it): "That doesn't seem very compatible with trying to make things work between us."
Unfortunately, this response went the wrong direction. Completely invalidating.
Technically, you are telling her that her actions (proving you are despicable) are wrong for the goal (trying to make things work). And that is completely true and completely correct.
But what she heard was that her feelings are wrong. Pure invalidation.
I leave and the texting starts. Clearly, she's upset, but I'm not about to validate her statements about me and ... .
Yup, it set her off... .and you are correct--there's nothing to do now but wait for this round of rage to burn off. No point trying to validate anything more NOW.
You got that part right. Cut your losses for now.