I did a small project for a new client who found me via Google and had also approached another programmer. The losing programmer asked my client who won the project and how much I charged. My client did not disclose this info. Is it ethical to ask who won and how much? In this case there was no bidding process - I work weekends while the other programmer didn’t reply until after completion.
There’s nothing unethical about this, it’s just basic research on their competition, perhaps why they lost, things they can improve on.
Would it be unethical if you tried to sell someone a used car, they bought one somewhere else and you asked them where and how much they paid?
Not in my books.
If you’re really afraid of clients disclosing that information you can add a clause to your contract stating they can’t disclose pricing information. Although I personally don’t think it’s anything to worry about.
I probably wouldn’t ask how much, but I would ask them (out of interest) what made them choose the other developer (so that I could take the answer into consideration in improving my own services). There’s nothing wrong with enquiring if there was a reason a potential client was “put off” your services. 
I like to do things like that, as long as it can be done gracefully. It’s a chance to show the lost client that you are professional (i.e. no hard feelings), that you care about customers (i.e. you are legitimately interested in knowing what drives them) and that you are a generally a nice person (i.e. one more chance to remind them how pleasant you are).
It’s a chance to wish them luck, to offer them the opportunity to come back to you if it doesn’t work out, and to leave on a positive note.
Agreed, In fact if you leave them with a good impression there’s a chance they may return to you on a future project if they were unhappy with the person they went to, or even if they had additional work which you could fulfil better than someone else, it’s all about leaving the door open.