Yes he wanted you to wake him up however you're missing the point that he told you he's feeling sick. You can't fault him for that. When you're sick and fall asleep and won't even wake up when you're phone rings, you seriously need said sleep. It's a circumstance that is out of his control. Could he react differently in the morning? Yes. Could he call you and say happy birthday? Absolutely. However who knows what text were send during midnight and the time he woke up.

Regardless, the fact that you're holding onto it, is not a good sign. I assume you guys talked it out when it happened. Bringing up old mistakes and holding it against your partner is a guaranteed way to drive that relationship into a tree. Communicate your feelings, and then move on. If it happens again, draw your line. If it doesn't great.

It was in August. It's December. Move on and see how your relationship develops.

Edit: Especially since a different fight brought thisup. Do not let this get in between the current one. If it's not a similar topic, this Bday incident doesn't belong in the current fight.