Don’t bother apologizing. If you need to communicate with him / her, just pretend they never sent the email or you never got it. Your old boss probably forgot all about it. If he asks, just tell him you got busy and it slipped through the cracks. I wouldn’t call it unprofessional to have not responded to his query. It happens all the time. What would be unprofessional is for them to make a big deal out of it or to make it personal.
You don’t owe anything to your former boss. If you did good work for them and helped them achieve their career goals, that’s all they can ask of you. If they went above and beyond the call of duty, send them a fruit basket for Christmas. Nothing more is asked for or needed.