I visited my branch and reported that funds were missing from my account. The banker took down my information and created a case file and told me that this would be investigated and that I would be contacted. Several weeks went by and Wells Fargo contacted me and told me that my request had been denied and that the money would not be returned to my account. In their investigation the bank found out that the wire transfer occurred in Texas. I am now out $1700, the bank knows I didn't withdraw the funds, and yet won't credit my account. Filing a lawsuit would be pointless since the amount wouldn't begin to cover legal fees.
Your state will have some state-level banking organization that regulates the business. On the federal level, the Office of the Comptroller of the Currency regulates banks. You can report this incident to either or both of those. Your state's attorney general may also be interested. The Consumer Financial Protection Bureau has an ombudsman office that could also be someone who could help you.
As the other comments have noted, you could always sue them in small claims court to return your money.
But, if I were in your shoes, my first stop would be at your local branch and asking for their manager to sit down and talk about this one-on-one. Right now, I think the biggest thing you need is someone inside their walls on your side. This conversation should be asking 'how did your investigator make that determination? why didn't they talk to me? what evidence do they have that this is legitimate? what kind of appeals are possible?' and so on. You need to not be angry here -- you will typically get much farther along with them if you act professionally and just firmly but politely insist that an error has been made and that you need their help to remedy it.
Your bank does not know you didn't withdraw the funds.
To be more specific it is not sure that you didn't authorize the withdrawal of the funds, or otherwise enable the withdrawal. You telling them you didn't is not enough.
One of the major problems banks have to contend with is "first party fraud". This is when someone asks another person to make a withdrawal from their account (sending them whatever information is necessary). Then after they have done it they go to their bank and say that they have been 'defrauded' out of the money and ask to have it refunded. If successful they split the money with the other person.
You don't say how the wire transfer was set up. If it was done online, and your credentials were used to do it, then you made an agreement that anything done with your credentials is to be treated as having been done by you. The bank may be using that agreement. To go against it you would have to conclusively prove that the bank was somehow negligent, or that your credentials were not used. If the transfer was set up in person then you may be able to show that signatures don't match or that you clearly were not present when they said you were.
Asking your bank "how they know that the transfer was not illegitimate" may not work. If they suspect that you were involve in the transaction they will not want to give away secrets of how they determined this.
You may be able to do this, but it's going to take work.