Generally, a person who has filed a lawsuit against another person may, through counsel, subpoena the business records of a third party within the jurisiction of the court, if what's in them might lead to the discovery of admissible evidence and the records are not otherwise privileged. If the third party is outside the jurisdiction of the court a more cumbersome process is involved, which involves invoking the assistance of a court in the other jurisdiction, usually a sister state.
Not legal advice, just general information. I don't practice law in California or hold California licensure. If you need legal advice, please consult a lawyer who does.
File a lawsuit against John Doe, prove that John Doe did something that you can due for and get damages, and then get a subpoena for the records. Expect Facebook to fight such a subpoena.
In the meantime, file a complaint with Facebook to get the profile taken down.
Mr. Goldstein is a Virginia-licensed attorney only. The information is not, nor is it intended to be, legal advice. You should consult an attorney for individual advice regarding your own situation. Answering this question does not in any way constitute legal representation. Contacting Mitchell Goldstein or the Goldstein Law Group does not constitute legal representation, nor is any information you provide protected by attorney-client privilege until otherwise advised.