I'm sorry it was so rough. I'm guessing T is not meaning to convey that the apology is not accepted. I can think of a number fo reasons a T mght not repond to an email like that.
1 - T might have not gotten the email
2 - T might be busy and not checked the email or had time to respond
3 - T might be waiting to bring it up in session

and I am sure there are any number of other reasons for T's lack of response.

I've had any number of phone calls to which T did not respond outside of session. A few were ones where I really felt rejected. It's always ended with T addressing it in session though. The biggest things were always things T wanted to make sure there was no misunderstanding about (which can happen for me both over written communication and ver the phone).

I'm guessing, unless you screamed at T, threw things at her, or something equally attacking, you have nothing to apologize for. We are in therapy for a reason: to work on things. It takes time, and sometimes it doesn't go exactly how others think it should go. That's ok...

Can you go to the next session and talk to T about it?