im sure he will call on Fathers day!
Actually, you should have her call him on Father's day and on his birthday... Maybe that will be enough to shock him into being a bit more communicative with your child. I doubt it, but what do you have to lose?
I know its hard to do, but you have to try and encourage the relationship with her father... she is only 6 now, but when she gets old enough to understand relationships better, she will remember what and what you did not do during these times.
Do you really want to do ANYTHING that would make her think you sabotaged her relationship with her father? Even if it is just the mere perception.. I know I wouldn't.