I haven't been able to find any laws on this. I'm not a lawyer though - but still, I was not able to find such a requirement. However you might be able to report them for cruelty to animals under Article 26 of the state code. The definition of torture includes allowing an animal to suffer unnecessarily, and torturing animals is prohibited.
Look under Agriculture & Markets, Article 26 (Animals).
http://public.leginfo.state.ny.us/menugetf.cgi?COMMONQUERY=LAWSI would certainly report these events to the state veterinary board. And possibly the better business bureau.
It's one thing if they don't have the expertise to treat birds and say so, but quite another if they imply that they know how to treat birds and then tell you that they won't see your sick bird for at least a month. Anyone competent to treat birds would have known the situation was urgent. Anyone with a heart, if unable to treat your bird, would have tried to help you find another vet who was able to.