It would be straightforward to implement a DICOM export plugin based on DCMTK’s stl2dcm example. However, I agree with @pieper in that this is a niche file format, therefore it is not clear if it is worth the effort of implementing and maintaining importers/exporters for it. I don’t think anyone form the Slicer developer team has ever seen such a DICOM file. It is also telling that DCMTK does not have a dcm2stl utility - most likely because people don’t come across such files.
Note that segmentations can also be stored in a mesh format, see: A.57 Surface Segmentation IOD. It is a bit more general-purpose representation (not so tightly linked to 3D printing as the encapsulated STL), but it has not become widely used either. I only remember this single discussion on this forum about this information object: Load DICOM Surface Segmentation Objects
If you just need to store encapsulated data files on a PACS in DICOM format then you may also consider Slicer’s embedded MRML scene format (read/written by the “Slicer Data Bundle” plugin). It is not compatible with any other software, but it is already implemented and it works with all data objects that Slicer supports.