It is James Dalton.
The name can be seen clearly in his medical record where it is written as Dalton James.
Below is the YouTube of the hospital scene from which the image is captured.
Wikipedia article of the film also says the same.
It is nowhere expllicitly said in the film that his first name is James. It is confirmed in the sequel Road House 2 and of course in the hospital scene.
A sequel, Road House 2, was released directly to DVD in July 2006. Set many years later and telling the story of Dalton's adult son, it featured no one from the original cast and only a few references to Dalton (who was reportedly shot dead before the film took place). The sequel confirmed that Dalton's first name was James, which could be seen momentarily on the medical chart in the original film's hospital scene, but which had been otherwise unstated.