Culturally, is it important that in a stich like this, that you (the H) must make amends and court the offended W?
If so, how important is this to you? How important is it for you to follow cultural edict?
While I like my FIL and MIL, they are not calling me and asking me to talk to my W to work things out. Although some of the things they are doing for my W is allowing me to be upset, because it feels like they are enabling her to walk away. Helping reduce her stress. It is contrary to what I know my MIL said to my W last year, which was to "give it time to work out". And my FIL and I had joked many times in the past about how it was now MY responsibility to cater to my W as the M allowed him to pass the buck, he has taken her under his wing again... it's what dad's do...
In truth, no one is suggesting that I should be seeking out my W, whether my in-laws (and their partners, my W's siblings, etc) my own family, nor my friends. While there are many who believe that M and the vows are sacred, it appears that everyone is respecting that this is between myself and my W to work out, or not, as we so choose, jointly.
I would submit that if your inlaws or any other come to you and ask you to try to work it out or involve yourself in your W's affairs, that you indicate the following (if it is how you feel):
"While I am still open to reconciliation, I am still working things out in my own life. I have let W know that I would consider working things out, but I can not force her to work things out."
That's it. I expect that by this time, you know that you cannot control her. By saying the above, you are simply affirming to yourself that you cannot and will not attempt to control her.
So unless there is some cultural protocol that you should be following and feel is important to you, then you are doing the right thing for you and the best you can do.