You did it wrong.
If you want to leverage more money from a job offer, you're meant to bring up to your employer that you have a job offer, and you're thinking of taking it because it offers more money.
That way, if they say "no", then they know you can take the job offer and leave.
What you're doing is probably not going to get you the raise, because you don't have the "threat" of leaving to back it up.
A lot of answers have stressed the "market value", but the textbook definition of "value" is what someone will pay for something - so if someone else has an offer (present tense) for you for more money, then that is your value.
If you don't have another offer though, then your value is what you're employer, roughly, is paying you.
I don't think you should mention the other offer, because you don't have it anymore. Companies don't mind paying more for people, and what you did is not considered "mercenary", but you did it wrong.
As one commenter mentioned, the fact that you didn't take the offer indicates that the company offers you something of value beyond the wage too, so whatever you negotiate has to take that into stead.
As to arguing your wage rise, you can push for 20%, no problem, that's just bargaining. To back it up, stress all the great things you've done, and how that is better than the average, and how if the average raise is 10%, well, because you're better than average you want 20.