Whilst I can understand your frustration, as a mother of 4, who's ex/baby father dropped off the map and has had nothing to do with them in over 4 years, and hasn't paid in any of that time either, I would be pleased if he showed any interest in any of his 6 kids. (2 to another ex) So I am reluctant to criticise your partner trying to communicate with his daughter, regardless how clumsy it may seem.
Although why were you reading his Facebook conversations?
It's obvious your unhappy, but it doesn't sound like he needs to change, since has zero incentive too. I am not sure there is anything else I can say. Maybe you should just do your bit and let him do or not do his bit until you are ready to decide you don't want to be in the situation anymore.

All the best for your future, hope you find what your looking for.