It is quite unacceptable for the other mother to call you and effectively blame your dd for her dd's mental health problems.
Girls fall out all of the time, it is what they do, but not every girl cuts her wrists due to this. It is not fair on your dd to be blamed for something so dreadful at 12 years old.
If it were me (and I have a dd this age) I would be advising my dd to keep her distance from this child from now on, being kind and civilised at all times, but not encouraging any deeper friendship so that she can no longer be blamed for the problems the other child is having.
Explain to your dd that this girl is in a bad place, but this is not her fault in any way and that the other girl will need extra support and help.
I would be putting in a call to the school to find out their views are on what is happening to be sure that there isn't a serious underlying bullying issue (You would need to address this quickly and robustly if you do uncover this) but given they haven't called you I would imagine there is not a big problem. Assuming it is just normal girls stuff, you will need an agreement with the school about their plans to support all the girls, and that you wish to be kept informed if anything changes.
Lastly call the mother back, and tell her you are very sorry that her dd is struggling so much, and that beyond a couple of very normal fall outs between the girls everything at school is within the range of normal pre teen girls as confirmed by the school. I would gently suggest she speaks to the school again regarding her dd, and seeks professional help. I would not be getting involved or allowing my dd to be held responsible for this.
Your dd can be not be the scapegoat here.