Someone recently asked me if he had a case against an equine surgery clinic that told his local vet during a telephone conversation to not send them the mare because they did not have room for her at the clinic.  The mare died 4 hours later of colic complications, and the owner stated that she would have lived if the vet clinic had admitted her and performed colic surgery.  The mare in that case was not a current patient of the clinic.  The owner would not have a valid claim against the clinic in that case.

The decision of whether to accept an animal as a patient is at the sole discretion of a veterinarian.  This rule is set forth in Article II.E. of the the Principles of Veterinary Medical Ethics of the American Veterinary Medical Association, which applies to all veterinarians in the United States.  The Texas Rules of Professional Conduct for veterinarians codifies that rule for vets practicing in Texas.  Therefore, even in emergency situations, vets do not have to take your horse if, for example, you cannot pay for the treatment or they simply do not have time to treat your horse.

For a vet to be potentially liable to a horse owner for injury or death of their horse, a veterinarian-client-patient relationship (VCPR) must first exist.  The VCPR is established when all of the following conditions are met:

1) the vet has assumed responsibility for making clinical judgments regarding the health of the horse and the need for medical treatment, and the owner has agreed to follow the vet’s instructions;
2) the vet has sufficient knowledge of the horse to initiate at least a preliminary diagnosis of the medical condition of the horse. This means that the veterinarian has recently seen and is personally acquainted with the keeping and care of the horse by virtue of an examination of the horse, or by medically appropriate and timely visits to the premises where the horse is kept; and
3) The vet is readily available, or has arranged for emergency coverage, for follow-up evaluation in the event of adverse reactions or the failure of the treatment regimen.
That said, Article II.F. of the Principles of Veterinary Medical Ethics does say that, in emergencies, vets have an ethical responsibility to provide essential services for animals when necessary to save life or relive suffering, subsequent to client agreement. This rule allows vets to arrange with colleagues to provide emergency services when they cannot be available to provide services. If a vet believes that they don’t have the experience or equipment to manage and treat certain emergencies in the best manner, the vet may offer to expedite referral to vets qualified to provide the emergency services.
This ethical rule does not rise to a "cause of action" against a vet clinic if you are not a current client and they cannot take your horse in an emergency. Further, the rule’s inclusion of the phrase "subsequent to client agreement" infers that vets may only have a duty to provide or help obtain emergency services for their clients (i.e. only after a VCPR is established).
Horse owners should keep handy the contact information of the closest emergency vets or clinics in their area and use them in a true emergency. If an emergency vet is not available, ask the veterinarian who normally treats your horse to give you a protocol of what to do in an emergency situation, and keep those instructions someplace you will be able to find them quickly in an emergency.