Australia's Constitution is very different from the American. It likely isn't the right place for such laws. (It requires a national referendum to make changes, it may predate independence, depending on how you date that).

However, there are other laws with constitutional significance. The Statute of Westminster, and the Australia Act.

But, most rights are guaranteed by Common Law, that is, any precedent or judicial opinion in any Commonwealth nation can be used for grounds that a right should be preserved when a decision is being examined by the High Court.

Constitutional amendments are very difficult to get up, the majority of attempts have failed. If either major political party opposes an amendment, it's essentially dead in the water. On current form the Coalition would drumbeat against it and Labor would do the bigger-badder-tougher-meanie-poo dance to try and neutralise the wedge.

Consider the fact that we are in the middle of an ongoing constitutional crisis, for which the political class knows an amendment is the only effective solution, but nobody in either major party wants to even try.