The site command isn't something that works incredibly well. Although it is a good way to do site specific searches and find duplicate content issues across your site.
One SEO, Olivier Duffez, asked on Twitter if Google can make the site command work by protocol. So you should be able to filter HTTPS vs HTTP results using the command. Gary Illyes from Google responded on Twitter saying, "we should revisit the idea."
Google actually did think about enabling the site command to support that at one point but they never ended up doing it.
Gary Illyes wrote on Twitter, "we were thinking about that at one point," but they never implemented it. He said he will revisit the idea and who knows, maybe it will become a reality?
Here is his tweet:
Forum discussion at Twitter.