Is there a one-click solution that builds a private organic group (OG) and invites another specified user to that group? I'm looking to let users initiate the creation a private organic group and invite another user to that group? I'd love to be able to craft a link (in a view of users perhaps) that adds a path like /request/new/og/{uid} to the routing table.

The idea is to match tutors to users needing help in a subject area. 

I starting from a View based on User Accounts that presents a table as follows. 

User       |  Tutoring Availability    | Request Tutor Group
-----------+---------------------------+---------------------
Abe        |  T/H Afternoons           | [ Request ]
Brian      |  M/W Mornings             | [ Booked ]
Courtney   |  Wknds Anytime            | [ Request ]
David      |  F All day                | [ Request ]
Mozzarella |  M All Day                | [ Request ]


In the request link above, I like to 


Create new private OG (of content type 'group')
Make the Logged in user the group manager
Invite (not automatically subscribe) the other user (potential tutor)


I'd like to not have the user see any node creation from as these private groups will have non-creative names such as "Tutoring Session for Rob and Jim on Organic Chemistry." Users will communicate via another content type that helps to track tutoring benchmarks. 

Update: I'd also be happy if someone can point me to a working code example of OG's API to create an organic group using functions from the og api. 

Bounty reward may require some screen shots to create a good reference point for the community.
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!