Approve public organic group
After having lots of fun with rules, I have found a solution. Whew! 

I created a rule:

Events: 
Before saving content

Conditions: 

Content is of type - Parameter: Content: [node], Content types: Community Group

AND

Data comparison - Parameter: Data to compare: [node:group-access], Data value: Public - listed in the...

Elements:

Set a data value - Parameter: Data: [node:group-access], Value: Private - accessible only...


I also send an email and show a message on the site, but that's not to important to the nitty gritty of the rule. 

The purpose of my rule is that when a user selects "public" for a group, that the group is created as "private" but a request is sent to a Group Admin to review and change to public once approved. In the meantime, the user can still add members and use the group in a private state.