Most likely, unless your mother suffered from some form of psychosis of course (which can be triggered) or a personality disorder (which can also be triggered), she wasn't "mad" either... Controlling parents/partners can actively cause depression of all sorts - which the abuser then uses as another "tool" on their abusive belt.
You do understand that your father may well have waited until your daughter was out of the room to have a go at you, OP, but that doesn't mean he's not hurting her. If he hurts you - even "just" emotionally - then she will suffer, too... and you cannot allow that to happen.
It's not up to the child to make a fuss of their parent(s) on their birthday - unless (a) they have a close relationship with them, and/or (b) it happens to be a milestone one. You know that 4 years ago, you didn't forget your father's birthday, that you sent him a text, but that doesn't feed into his ego and he's using that text as a stick to beat you with. Who is he displaying his cards to, I wonder; who is he bragging about "how much [his] family love [him]" to? Because, if he's anything like my daughter (who has psychosis as well as a diagnosis of NPD), it's going to be someone. And until you satisfy that need in him for being the centre of attention, he is going to verbally and emotionally pick away at you until you're left feeling that he's right, you're wrong, up is down, and white is gold. That's what you're teaching your daughter. That her grandfather has the right to make her mother feel like shit on his shoe. If it's not this, it'll be something else that he's attacking you over... and believe me, for your own sanity, OP, and the emotional welfare of your own impressionable child... you have to stop allowing him to do this to you.
You say that your daughter isn't close to him (good!) and that he favours your brother/his children over you/your daughter. So you're also teaching her that daughters are worth less than sons. And even though you might not be aware of this, she'll be drinking it all in and absorbing this horrific lesson - her mum doesn't understand that she's worth more than abuse.
I know it's difficult. My mother and daughter are both cut from the same cloth as your father seems to be - and for years, I thought that they had the right to abuse me every which way. It was some very wise posters on here who set me straight and, y'know, life's been better since I imposed boundaries and refused to bend them. As I'm typing this, my daughter (who's 23) is currently having some sort of temper tantrum about her computer not working. I made a suggestion that she could use mine, if necessary (she has an interview today and "all my stuff for it" is on her refusing-to-work computer) and she started yelling at me. I simply shrugged my shoulders and pointed out that when you talk to people like they're shit, they feel less inclined to go out of their way to help you.
When your father passive-aggressively demands things from/of you, you feel disinclined to give him what he wants - be that a birthday card, or anything else.
Give it a try, OP, you've got nothing left to lose. 
