As I have no experience with this type of thing at all, and you have not indicated the age of your child, I can only really give you my gut reaction as a mother of 3 adult children - and I must add that I made far too many mistakes as a mother when they were younger, 2 of which I now consider to be very serious mistakes.
My gut reaction is that if your child's father does not want anything more to do with his child, and actually makes disgusting comments about your child, then I think that trying to keep in touch with your ex, or even just sending him a card from your child, would be actually stamping on your child's self-esteem. I am going to assume for purposes of ease of explanation that your child is about 6 years old, so old enough to remember their dad, but no-where near old enough to explain everything to them.
IMO if you encourage your child to send a Father's Day card to this nasty man, it is like you are saying to your child when they look back in retrospect when they are older
"This horrible and selfish man is your father and even though he does nothing positive for you, but instead is very negative towards you, that is all you are worth, so show him respect (and love?) by sending him a card, it doesn't matter that he has not acted like your father (biologically or otherwise), as that is his legal position, and therefore deserves the benefits due to a loving parent". Even though your 6 year old wouldn't understand all the nuances of that last sentence, they can "feel" even from a much younger age, when someone loves and cares for them.
So to my mind, your little one would be thinking/feeling (even if not properly understanding that feeling), I cannot be worth much to either my mum or my dad if they think I should stay in touch (and even show affection to) someone who wants nothing to do with me, and in fact is emotionally abusive to me. So no, I wouldn't be getting my child to send that card, and if they were asking questions about why they weren't making a card for their dad when everyone else at school was making them for their dads', I would be asking opinions from child psychology experts, friends who are obviously good parents, and yes, here on MumsNet too -as long as you feel able to seperate the wheat from the chaff (and sorry, I don't know which one I am) - on how to answer those questions; I might try to put more emphasis on reminding your child about all the people who do love him/her, starting with you, your parents, and your ex's parents (if they are lovely people), any aunts, uncles, cousins, friends etc and tell your child how they are a great, kind, caring, loving person, who deserves all the love that is given to him/her, and maybe be a bit vague on why your ex seems unable to love anyone but himself - and maybe not even himself, which is why I mentioned if his parents were lovely, because maybe there is some history there?
I think that your mothering instincts are spot on, and I truly believe that your child is going to thrive despite the waste of space their father is, because you are so obviously such a loving and caring mum, who also has the intelligence to question things. Of course 2 loving, kind parents who love and respect each other as well as their child, who have commonsence and some intelligence are the ideal, but many of us cannot give that to our children, so we do our best, and I think that your best OP is going to be brilliant. I hope you get the help you need from here and other quarters 💐