I think the underlying point is that you are not going to be able to shield your child from the wagging tongues of others concerning his actions. People can - and will - talk, and it is almost certain that most of his peers already know about his situation.
Certainly, you can complain to the school about the teacher speaking "out of school", so to speak, but it is doubtful it will result in anything more than an admonishment from the administration - if that. Hopefully, that is all that you expect.
Teachers are people, and people talk. Whether she should have said anything or not is a matter of individual opinion, but there is almost certainly no legal requirement for her to have stayed her tongue.
I hope your child has learned his lesson. But, consider that part of this "lesson" can be the humiliation that comes from the act. Shielding someone from shame and embarrassment serves no purpose except to, perhaps, encourage further such acts. The message from home, and from the authorities, needs to be consistent. That message needs to be that theft is wrong - even more wrong that a teacher passing the embarrassing news on to someone they perhaps should not have.
Ultimately, our children will learn from us. I choose to hold my children's feet to the fire. I hope that you have done the same.
- Carl