Can I share a geoprocessing service (export web map) from either ArcMap 10.4 or Pro 2.0 to Portal 10.5.1 as a Hosted Service? I can share it to ArcServer and share it with My Content in Portal but is it possible to publish it directly to my hosted services?
A hosted service really only applies to a hosted feature service. You'd publish your GP service to a federated Server, or the hosting Server, and it's automatically added to your My Content and can use the sharing settings within Portal to control security.