You know tsar, I was rethinking my original answer. I answered it based on the assumption the OP is as s/he described, and that the starer was as s/he presented.
When I catch someone staring, there's an automatic instantaneous checklist that goes through my mind.
Written here in no particular order, I ascertain if....
The person appears aggressive. There are times when it would be safest to avoid connecting with the person. They might be drunk and volatile, they might be just jonesing for a confrontation. It's all dependant on the situation.
If they are staring, or just "lost in thought". Ususally someone like this isn't right on top of you, they may be sitting across the room and not even realize until eye contact is given back that they were what appeared to be staring.
There are few situations that I would give them the Bobby DeNiro treatment. More like me would be to say "Did you need something/want to say something?" etc.
Even more like I would look back at them and furrow my brows at them to let them know they were staring and it wasn't welcome.
Wally and I were in a restaurant a few weeks ago, and suffice it to say, we were both really unhappy with everything there. I'm sure the looks on our faces showed that. When I looked around at one point, I saw a man a couple of tables away, blatantly staring, not like he was concerned, but almost like we shouldn't be expressing our feelings w/ each other. I saw him, looked away and immediatley back, thinking he might have just been glancing over. He wasn't, he was still staring. So I gave him the full on "Whaddayoulookinatbub?" scowl, and he burst into flames.
Good thing Wally didn't stare at him.
Admiring or interested glances are welcome, staring is not. I get into interesting conversations with people all the time because we glanced at each other and start talking. Someone staring is more apt to make me leave the room.