I was having this discussion with my wife the other day. She pointed out that there are lots of landscape structures - many of Capability Brown's parks, for example, that are protected by English Heritage. Come to that, there are plenty of golf courses - the nine holer at Stowe, Tom Mackenzie's new course at Heythrop in Oxfordshire, the course at Luton Hoo in Bedfordshire, or Rudding Park in Yorkshire - that are within listed parks.
Now, based on this, I can't see a fundamental reason why someone should not apply to have a golf course listed in this way. EIGCA produced a document, in association with English Heritage, a few years ago, entitled 'Golf Courses as Historic Designed Landscapes'. Anyone can apply for a building or a structure to be listed. It would be interesting to see how English Heritage would respond if someone did this for a golf course.