...customer was staring at you? or harassed you, and the manager said well what could he do about it? Would you walk out without paying or what? in the UK
If you ate the food, you are legally obligated to pay. You asked for a service, and they provided it. I wouldn't go back there again.
As far as the person staring, if you complained to the manager, and he did nothing, why didn't you do something? I would have, very loudly, asked the guy why he was staring.
There is a difference between be looked at and being harassed. If the customer is harrassing me and fails to stop when I ask him to, then I would expect the manager to act on my behalf if I asked him to. If he failed to do so, I would feel justified in walking out - possibly without paying, depending on how much of the meal I had eaten.
If the other customer was simply looking at me from his table, I would pay for my meal because it is no fault of the restaurant that I am unhappy about being looked at. The problem is between you and the other customer. Provided the other customer is conducting himself in an orderly manner, there is nothing the restaurant manager can or should do on your behalf. If you feel strongly about it, what is to stop you talking to the other customer?
For starters, you must have been staring to know that the other customer was staring at you. A quick glance is one thing, but a stare takes time - both ways. You are also guilty. It could very well be that the customer is deep in thought and is unaware he is even looking at you. It may also be that the other customer has failing vision and unaware that his gaze has you in the line of sight as you suspect. My advice to you is to stop staring back at the customer or reposition your seating so that your back is turned. I can't imagine what you expect the manager to do for your own sensitivity.
Grow up and handle your own insecurity by diverting your own eyes. It is not the manager's responsibility to tell customers in which direction they can cast their gaze.