The key thing for your daughter in times like this is that she feels that both of her parents are around for her. Making demands and frankly being petty (as some on here are suggesting) is not in her interests.
The best outcome for any child in these situations is maintaining a loving relationship with both parents.
No, it isn't. The key thing, as would be considered by law is the welfare of the child. Having a relationship with both parents is considered as being in her interests unless it is damaging.
One of the ways it may be damaging to maintain contact with an absent parent is if the parent is not building the contact around a child for example by insisting the child spends contact with a new gf straight away.
It is in the interests of the child that her environment is made stable and that the transition to new circumstances is made smooth.
A parent who has left the family home needs to re-establish their relationship with the children by committing to regular and stable contact with the child with a view to making the child feel secure in the fact that they have left the home and not them.
Introducing a new gf because you want to prove the relationship is serious enough to have left the family for is a common mistake which damages children and primarily damages the child's relationship with the parent who has left.
You should absolutely be insisting that your ex prioritise your dc, this is in his interest as well as his daughter's, if that means making demands in order to protect her from being used to salve his guilt then so be it IMO.
People are incorrectly staying the law.
Constitutional principle states that which is not forbidden is allowed. With no court order there are no restrictions on what you can do. If you act in the best interests of dc and can back that up with strong arguments then if it goes to court you will likely be listened to.
He should see his dc but it should not be with his new gf until their relationship is stable and secure.