From that day she won’t / wouldn’t talk to me.
It refers to some day in the past. What is the difference in meaning between two ways? Compare it with “From that day she doesn’t talk to me”?
Let me guess that both ways are appropriate, but won’t implies that she still doesn’t talk to me now, at the time of utterance.
If you want to talk about the past, use wouldn’t talk.
From that day she wouldn’t talk to me.
This sentence refers to the past and tells nothing about the present or future. Something happened on some day in the past that caused her to refuse to talk to me since – that’s all we know from this sentence. It is likely to be found in stories that use the past as a narrative tense.
If you want to talk about the present, use hasn’t talked, not doesn’t talk.
From that day on she hasn’t talked to me.
You have to use Present Perfect, not Present Simple, in this sentence if you want to say that it is still true now, because you are defining a time frame by including from that day. If you omitted from that day, then the sentence “She doesn’t talk to me” would sound totally fine.
If you use won’t talk in your sentence, then it will refer to the future:
From that day she won’t talk to me.
This sentence specifically refers to the future. It has nothing to do with the present. It hasn’t happened yet. It will only happen on some day in the future and from then on, she will not speak to you.