First of all, UL does not "approve" anything. They list some things, and they label some things, but they approve nothing. As to the NEC's requirements, try looking at 90.7. It is not as clear as one could wish, and that may be by deliberate intent. What it tells me is that a laboratory's testing program is a good basis that the AHJ can use for approving a component that is part of an installation. Having a UL listing takes away some of the burden from the AHJ's inspection process, in that the AHJ does not need to repeat any testing that the UL will have already conducted.
I seem to recall that there is at least one place in the NEC (and I don't have an electronic version to look for it) that explicitly states that that particular item must be UL listed. But that is not a generally applicable requirement.