I want to take a sick day for myself but not take one for my wife
Why don't you want to take a sick day for her?
I am fed up with her sometimes.
Couples can regularly get into conflicts, but it doesn't mean you should neglect her. She is your partner in sickness or in health, so the both of you have to take care of each other.
She should also support me too.
Well I'm sure she does when you are sick. But I think as her husband you should always take care of her when she's sick as well.