No, you can't. The correct way to refer to the week after this one is "next week" . The correct way to refer to the day after this one is "tomorrow". If you want to refer to the day following another day, you say " the next day". For example, "He left school at the end of June and started work the next day."

"My birthday is tomorrow " You can say "my birthday is next week" you can say " my birthday is the next day " /"my birthday is the day after" but only in a particular context, ie: "I'm leaving December 30. And my b-day's the next day"

"My birthday is the next day" can be said as "My birthday is tomorrow" If you want to say that your birthday is next week you can say "My birthday will be next week" c: