Sure you can. Skype did it, Hitachi¹ did it.
¹http://www.pgcon.org/2008/schedule/events/57.en.html
> properly handle JSON right now
Well it has V8 running on it.
> It also has a fixed schema
You don't need to use it like that if you don't want to.
And again V8 is nice and all but it is rather bolted on as opposed to something that is was built from scratch to support JSON. This shows in terms of feature support and most importantly ease of use. I can't just annotate HashMaps, Lists etc in my Java classes and have them serialized to PostgreSQL in JSON format.
I am just saying that PostgreSQL is a great and all but it is not a proper JSON document store style database anymore than hacking SQL on top of MonogDB would make it a RDBMS.