No, of course, he hadn't: that's not the way the Westminster system works. He'd been elected by his electorate to be an MP, and he was elected by his party to be party leader, and he'd been appointed by the monarch to be PM. Obama was elected as leader by the people, but that doesn't happen in parliamentary systems.
When Larvatus Prodeo complains "Much as those attached to the verities of the Westminster system might protest otherwise, it’s difficult for many to come to terms with the fact that an elected PM has been torn down" he's getting it out of kilter. The Australian system has certainly developed into something where the PM has something of an independent power base by virtue of election, but we've just been reminded of how little this is reflected in the actual rules.