The Commonwealth shall not make any law for establishing any religion, or for imposing any religious observance, or for prohibiting the free exercise of any religion, and no religious test shall be required as a qualification for any office or public trust under the Commonwealth.
It should look familiar - it was largely copied from the US Constitution. Australia does have separation of church and state.
Forget Australia. In the USA there will be atheists that will file lawsuits about religious names for hospitals if health care is 100% run by the government with taxpayer money.
You know that will happen.