Most of the hundreds of birds and more than 3,500 horses that enter the United States at Kennedy every year are still put onto trailers for the 2-½-hour ride to Newburgh, even though the cargo planes could practically pull alongside the Ark, where the horses could be transferred to a dolly and wheeled into a stall moments after landing.

“I watch them go by my door every day,” Mr. Cuticelli said, “trucked up to Newburgh.”

“It makes no sense to me why the Port Authority, the U.S.D.A. and all the other agencies involved in the creation of this facility are not supporting its use,” Mr. Cuticelli said. “I entered into a technically, legally binding contract with exclusivity. Otherwise, I never would have done it.”

The Port Authority and the Department of Agriculture declined to discuss the lawsuit.

Joelle Hayden, a spokeswoman for the U.S.D.A., did say that her agency “does not mandate use of any one import quarantine facility, airline or animal handling service. We support the use of all approved entities and services.”

And a spokesman for the Port Authority said in a statement that the “limited exclusivity provision enables the private veterinary clinic to provide the first-class facility and services that J.F.K. requested.”

Critics, mostly shipping agents and transporters, said that the privately owned Ark was too expensive and unnecessary, given the existence of the Newburgh operation.

But Dr. Scarlette Gotwals, a veterinarian and director of flight operations for Horse America, a shipping agent, said she was mystified as to why Ark has not been embraced by the horse industry.