Quick background... I'm Mom, share two kids with my ex (Son 19 next week, daughter 16). Divorced 12 years. He's remarried. I'm not. Our divorce wasn't friendly, although we found a middle ground about 6 mos out and were able to work together reasonably well (not perfectly, but reasonably well) for ~3 1/2, 4 years. Things kind of went to heck in a handbasket when he met his wife. I won't go into the details as it would take much too long, but suffice it to say that there is plenty of blame on all sides. I'll own mine, but will add that I have made numerous attempts over the years to try to make amends for my part of things.
As the kids have gotten older, their relationships with their Dad have deteriorated. Our son's has never been great. He's attempted to breach the gap, but so far nothing's worked. He expresses his frustration through his music.
Our daughter has always had a great relationship with her Dad. Until he forgot her 16th birthday. He didn't acknowledge her 15th, either, but she let it slide figuring that he was busy and just forgot. (I will note that his stepdaughter's bday is 4 days after our girl's - and he got on both of ours for only sending her an email and not calling the year of daughter's 15th). Her Sweet 16 came and went. No card. No call. No email. Nothing. I threw a party for her 2 months later (better weather) and invited her Dad - told him it would mean a lot to her to have him there, etc. The response I got was "I won't be there." I never told her.
He did come down for our son's graduation from HS and took them out (originally, only our son and a friend were invited - she was not. Our son told him that he would not go if she wasn't included). So, the invitation was amended.
She used to call him weekly, always got the answering machine, and no return call. So she stopped. He sent her an email that it was clear she wasn't interested in being part of her family in (State) so he'd respect that. They are now at an impasse. She will not contact him because she feels it's time for him to step up to the plate (there is obviously a lot more to the story - I'm providing a skeleton).
I told her last month that she could use email, phone or text, but she would wish her Dad a happy birthday. She came back with "He didn't wish me one!" I told her that I know he didn't, and I know it hurts. But she should not be that person. In the end, she did text him. with no response.
For better or worse... I emailed his wife. Despite our differences, we're mothers and I figured she'd understand how hurtful it would be to a girl for her Dad to ignore her birthday. I said that despite our issues, I really wanted to see both kids to have a good relationship with their Dad, and the current issues between #2 and Dad pained me. That #2 was upset that her bday was ignored two years in a row, and one of those was a "big" bday. The response I received was "If #2 has a problem with us, she can talk to us about it."