Your friend told you that she didn’t want to talk to you again and to leave her alone because she was hurt by the messages that she received from you when you were drunk.
That notwithstanding, your friendship with her is not necessarily over.
However, even though it has been a month since she told you to leave her alone, you should still keep your distance.
She may still be really upset over what happened.
You should stop thinking about trying to make things right.
Just let her continue having her space.
Focus more so on making sure that you are a positive influence around mutual friends or acquaintances that the both of you may know.
Be extra helpful to them.
This way, they may continuously report back to her about how much of a good guy you are and this alone may eventually make her feel as though she should give you a second chance.