I had a similar experience, although I did not see the dialog you posted. In my case it was reported as an ssh failure. Like you, I could do a git push from the command line but Xcode was failing.
To get it working I edited ~/.ssh/known_hosts and deleted all entries to do with my git server.
I then manually logged into the git server (via ssh) and when prompted accepted the connection, adding the keys to known_hosts.
Following that Xcode "push" was working again.
I have no idea as to why the failure occurred but I am happy that it is working now.
Topic:
Developer Tools & Services
SubTopic:
Xcode