Dr. Bernie Moore is honored here, perhaps because he served as Chairman of the Board of Trustees for two decades. He was actually one of three physicians who founded the hospital. The other two were Drs. Ellis Sparks and James Earl Robertson. According to the Wikipedia stub on Crestwood, businessman Archie Hill was also involved. The facility opened as a nursing home in 1964, and became a hospital the following year.
That Wikipedia piece says the hospital was named after the "district" of Huntsville in which it's located; I have yet to find any more information on the claim. My mother has lived in Huntsville since the early 1950's, and she's told me she doesn't remember Crestwood as a neighborhood or area of the city.
Oddly, the hospital's web site linked above has no historical background that I could find. You never know when a little history will pop up somewhere, though.