Is there a one-click solution that builds a private organic group (OG) and invites another specified user to that group?
I would use Page Manager (part of ctools).

You can add an action or a tab to just about any page so you could have your view of users with a tab or link at the top (depending on your theme/styling) that requests a new organic group.

The URL would be based on the user view.  For instance if your user view was at /people/users you could add a path at /people/users/addgroup which redirects to the OG group creation page.

Alternatively you can create whatever URL you want like /request/new/org/id and give people the link for it... then use that URL to redirect to the OG group creation page.

In order to hide the user being added to the form you should be able to use a parameter in the URL like request/%tutor, then pick up the argument in Page Manager and auto-populate the user as a member.

As for automatically filling out the form?  Well, Automatic Nodetitles has worked well for me, you can build titles out of available tokens which is pretty neat.  And you could probably run some javascript to click the submit button on page load but that's a kludge.

If you don't want them to see the node creation screen then yes, the API is the way to go.  I haven't used OG so I'm not familiar with it, but most contrib modules have a .api file which gives an explanation of the available hooks.  But you probably want to use one of their functions, just look through the code until you find one called og_create_group or something like that, figure out what to send it as parameters and you're off!