I agree with ImMissHannigan, you are getting a hard time here OP.
I hope you can encourage your son to get some help for his depression.
Personally, I would not follow my son's instructions. I would be open and up front about the fact I wanted to make contact with the mum, apologuise for whatever has happened before now, and see if I could have a supportive and friendly role in my third grand child's life.
Hopefully, in time, your son will see that what he is doing and will change his mind. I always think it is very sad to see much older men trying to make up for their massive failings and trying to forge relationships between themselves and their children.
I do understand he did not want to be a dad again, he has a financial obligation and I hope, sincerely, he will see that he has a moral obligation to the child he helped to create.
A male family member had a baby with a woman (a long time ago) and then said he did not want any more. As far as I can make out she went 'behind his back' and got pregnant again. It broke up their relationship.
However, he has stuck by the kids and ironically has ended up with more children with his new partner.
I really hope if you can get your son some help with his depression you might see a change in his attitude. But whatever happens you can show him the way to be supportive and work towards a good relationship with your grandchild. In the long run this will be the most supportive thing you can do for your son and your grandchild, IMHO.