In a situation like this, I’d guess that there are a small number of people for whom it would be better to actually keep this type of relationship going. She called your wife a pig!!! That is unbelievably disrespectful! A really good apologist with a strong faith and a strong sense of self-esteem MIGHT be able to have a positive effect on her - but only IF she agreed to never resort to name calling and being so disrespectful. If you listen to Catholic radio when a person calls in and just gets insulting, you’ll notice that even the CA apologists don’t just take it - they insist on a certain level of civility.
But here’s the thing - your said your wife was crying all night over this. I don’t know anything about her apologetic skills, but this is a clear signal that she is in too deep emotionally. As rediculous as she knows her friends’ opinions are, she is very personally hurt by this, and IMHO it wouldn’t be wise for her to subject herself to those kinds of abusive insults. As another person who can not just let insults roll off my back, I can tell you sometimes you need to protect yourself - sometimes you are not in the right positition to correct the sinner. And sometimes the best thing you can do for the other person is to not accept their misbehavior. A letter written that clearly states that you will not tolerate the kind of abusive/disrespectful/dehumanizing treatment of yourself may make this person take pause, more than if you nicely tried to argue about overpopulation with her. (BTW, this woman’s obsession with overpopluation just shows that she doesn’t understand the value of human life, and showing her that your wife’s life is valuable enough not be be riduculed may be the first step. She won’t be ready to listen to overpopulation arguments until she realizes that human life is precious and not worth comparing to pigs.)
I also don’t know about the other issues between you and your dw that you referred to, so I don’t feel comfortable advising you on how involved you should be. But as a wife who has a dh with strong opinions, I know that it is very important to communicate your concern for her in a loving way. Instead of saying, “You should cut her out of your life. Period.” say, “I’m really concerned for you. I think she is being emotionally abusive to you. I hate the way she’s making you feel. You deserve to feel better than this. etc.” You have to let her know that you don’t think she’s crazy for wanting (at least partly) to continue the relationship, and that you understand that this is her decision to make, and you just want to be supportive of her."