No, to my best knowledge, private to public network communication cannot be done with Oraclize since the Oraclize service is only listening for events on the public Ethereum network (and some public test-networks like Robston, etc.). Perhaps, if your project is big enough, you could contact the Oraclize team and ask for them to include your network.
On the bright side, however, this whole process should work in the other direction, having a transaction on the main Ethereum network trigger an action on any other network. What you would have to do here is use a computation query in your Ethereum smart contract to execute a dockerfile which in turn executes a JS file, in your case using web3 to call a function on your private network. (Provided, the executor of the query-computation has access to the private network)
http://docs.oraclize.it/#integrations-private-ethereum-based-chains. Oraclize just update their docs page and added a section for using oraclize in private chains.