You could, in theory, file an appeal, in state court, alleging a Constitutional violation, or, in theory, file a new separate case in Federal court. THe chance of success in either one is tiny. Just because your attorney ARGUED that your constitutional rights were violated doesn't mean that any judge in state or Federal court is likely to agree with him; one judge has already disagreed with him.
Of course you can sue. Anyone can sue for anything.
What you really want to know is do you have chance of winning.
No way we can tell, since you posted no facts or circumstances to assess. Nothing you posted gives rise to any violation of equal protection—you made the declaration that your rights were violated, but you have to be able to state exactly what occurred and why it was a violation.
Losing a hearing in family court--which at bottom is the basis of your claim--is hardly grounds for an equal protection claim.
Not sure I recall any particular Constitutional rights that fathers have over and above mothers, and it appears that one judge already decided your lawyer's argument didn't hold water--that doesn't mean the decision was correct--but it does give an indication of how things might proceed downstream.
Since you already have a lawyer, why don't you ask him/her about next steps regarding a federal claim?
NOT LEGAL ADVICE. FOR EDUCATION AND INFORMATION ONLY. Mr. Rafter is licensed to practice in the Commonwealth of Virginia and the US Federal Courts in Virginia. There is no implied or actual attorney-client relationship arising from this education exchange. You should speak with an attorney licensed in your state, to whom you have provided all the facts before you take steps that may impact your legal rights. Mr. Rafter is under no obligation to answer subsequent emails or phone calls related to this matter.
I do not see anything in your post that would permit or be a basis for suing the state of TN. You disagree with and are upset with the judge. The judge is immune from suit in this circumstance. Your avenue to redress your claim is an appeal, timely filed.
The above is not intended as legal advice. The response does not constitute the creation of an attorney client relationship as this forum does not provide for a confidential communication.