I'm sorry to say that I think, overall, it will probably make dating a little more difficult for you. But that is not so short that you should despair over it. I would date a guy who was 5'1". I think many women ideally would prefer to date a man who is taller than them, but in practice can be much more open about height range. Of course, there will be women who simply won't date a short man--don't worry about them. There are plenty who would. It will probably help if you do not seem like you feel your height to be inadequate.
This is assuming you don't grow any more, of course. I don't know why the doctor would say that you won't. I'm pretty sure every one of my male friends kept growing into their late teens, one or two even until they were about 20-21. Women usually stop growing in their mid-teens, but men generally keep growing for a few more years. But even if you don't, I wouldn't worry about it too much.
_________________
"A flower falls, even though we love it; and a weed grows, even though we do not love it."