How to implement Schema.org on HTTPS pages?

tl;dr: Use http URIs.

In this answer on Webmasters SE I explained why you should favor http over https Schema.org URIs: The http URIs seem to be canonical, as the actual definition of the Schema.org vocabulary only defines http, not https. In addition: all examples (even on HTTPS) use the HTTP variant, the authors mentioned that they prefer to see the use of the HTTP variant, and RDFa’s Initial Context defines the HTTP variant only (so most of the RDF world will use HTTP).

In this answer on Webmasters SE I explained why you should not use protocol-relative URIs for vocabularies: Vocabulary URIs typically don’t get dereferenced, and there will never get something embedded from a vocabulary, so there is absolutely no need to use HTTPS for these just because you use HTTPS (it’s similar to simply linking to an external page, which might not even be accessible via HTTPS). On top of that, your Schema.org markup would no longer work if the document is accessed via a different protocol than HTTP/HTTPS, and it’s likely that some parsers won’t be able to recognize that you are using the Schema.org vocabulary because they might look for full URIs without applying URI resolution for the itemtype attribute.

There's been an update to that answer on Webmasters SE (dated November 2015), with a link to the schema.org FAQ about https:

Q: Should we write https://schema.org or http://schema.org in our markup?

The short of it is that schema.org will be moving to https, and you can use https URLs now, but there's no rush to switch.