Delta's full statement is here.
In short, while Delta can't completely ban emotional support animals, it's going to require that they be registered. As well as the "emotional Support/Psychiatric Service Animal Request form which requires a letter prepared and signed by a doctor or licensed mental health professional", the passenger must provide.
1. a signed Veterinary Health Form and/or an immunization record (current within one year of the travel date).
2. a "signed Confirmation of Animal Training form"
All these must be provided to Delta’s Service Animal Support Desk via Delta.com at least 48 hours in advance of travel.
I guess this is going to clamp down on pax just showing up at the airport with their animal and a piece of paper signed by any old "doctor".