I think I found the Xcode way to push a existing project to GitHub or any other remote repository!
Preconditions
- I assume that you have a project created by Xcode 4 with a local git repository.
- You created an empty remote git repository (I call it RemoteRepository in in the following example)
Push it to remote
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Open the Organizer in Xcode an go to the Repositories tab
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At first click on Remotes and then on “Add Remote”

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Enter a Remote Name and the Location of your remote repository and click on create

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Then select your project and open
File -> Source Control -> Push ...
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Select your remote repository and wait until it is connectable! Look for the Create in the brackets.

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Now your old project is published to GitHub

I hope it will works for you also! In my case it works with every remote git repository.
Please contact me on twitter if something unclear or wrote in bad English!