The employee should absolutely NOT call the customer. Was it an honest mistake, i.e. did the employee KNOW that this wasn't the account holder making the deposit? If not, and the default is to print out the balance on the deposit confirmation, then the employee, IMHO did nothing wrong.
And quite honestly, even if they did know it wasn't the customer, it doesn't sound like it was intentional. The employee did her job, and it probably didn't even cross her mind that the person making the deposit woudl then SEE the account holder's balance. I know if it were me, if I were busy doing my job, that's not something that would cross my radar. Esp if printing the balance was the norm, and nothing special had to be done to do that. It seems to be just an inadvertant oversight on the part of the employee. Yes, privacy rules are in place for a reason, but it seems to be one of those situations that you don't do anything about, or know to do anything about, until it actually happens.
The manager absolutely should placate the customer, and NOT get the employee involved. that will just open up a huge can of worms. Esp if all the customer wants to do is yell at the employee. Its over and done with, and hopefully, won't happen again. But to dwell on it is both unprofessional, and not nice to the employee.
I posted in the SS thread about a customer I had the other day who told me I was rude, i.e. I didn't give her the instantaneous answer she wanted, beacuse I cannot access that specific information. But I was most certainly not rude. Manager got it, called her back, but had to leave a message. As far as I'm concerned, the matter is closed. However, if the customer then made a stink, and I was ordered to call and apologize, I would refuse. I was nothing but polite and professional, and she was the rude one. So in my mind, there would be no reason at all for me to call the customer back, esp since she too did not let me get a word in, after she launched into her diatribe.