I can't really comment on the what should be done in the US because as I've said, I don't know nearly enough to put forward any kind of opinion that is both meaningful and specific enough to be useful. I vaguely imagine that similar approaches to what we should be doing in Australia would work.
1. I've never heard anybody seriously suggest or ask that white people leave cities or the lands that they're living on in Australia. That's not a solution and nobody thinks it is. At the moment the land rights movement in Australia, which has been very successful, is aimed at giving the land indigenous people currently live on to them if they've been living on it for many generations. This has now almost completely occurred (although only in the last few decades).
While issues like "justice" are important, pragmatic responses and recognising reality is also important. The reality is that the indigenous Australian population is only a tiny fraction of what it once was, so vast areas of land aren't necessary to be given back to them, nor would this be helpful as they wouldn't have the resources to manage too much land and would inevitably sell the land on. Land rights is essentially rights for a group of people, no specific individuals, so giving land to people in the context of land rights that they then sell raises significant ethical concerns. The other issue is that in Australia indigenous people have significantly lower health, wealth and education, especially those living in rural communities. Worrying about "justice" for wrongs that occurred generations ago should be less of a priority than improving the current situation. Of course that's not to say that "justice" is not an issue that needs to be addressed, and especially in issues that occurred in more recent times (such as the so called "stolen generation" and policies of cultural assimilation), and of course we can address multiple issues, but some level of prioritisation needs to occur.
2. The underlying issues facing indigenous people in Australia, especially (but not exclusively) in rural communities are complex and there are clearly no easy solutions. Indigenous Australians, like many indigenous groups around the world, have far lower life expectancies, higher infant mortality, high levels of drug dependencies, high prevalence of domestic violence, high prevalence of child sexual abuse (in a vicious cycle where around 80% of offenders were abused as children), disproportionate representation in prison etc. To expect some kind of simple solution to these issues is naive.
3. However there are definitely some under lying principles that have been learned over the years of trying to address these problems that will need to be part of the solution. They are as follows.
*Self determination and empowerment
Indigenous Australians need to be in control of their own destiny and feel like they are. This doesn't necessarily need to be run at a federal level, or even a state level - well resourced community governments have proven so far to be the most effective method for dealing with the issues facing indigenous communities.
*Encouraging education
The difference in socio-economic demographics between indigenous Australians and their levels of education is a clear indication of how important education is as a long term solution to the immediate problems these communities face. More resources need to be allocated towards this and cost/benefit analysis seems to indicate that taking that funding from almost anywhere else in indigenous management would still have a long term positive effect (obviously with in reason). Over the long term, significant investment in indigenous education, especially for rural communities would actually save the government significant amounts of money and would help generate wealth and industry in communities that are largely welfare dependent.
*Legal equality
In the eyes of the law, individual indigenous Australians need to be treated in the same manner as other Australians. This specifically means giving penalties for abhorrent criminal behaviour the same kind of punishment it would receive if carried out by a wealthy whites in cities. Giving lesser punishments for terrible sexual assaults on children on the claim that such behaviour is "ingrained within the culture of those communities" sends a terrible message to everybody and does very little to address such issues. Yes definitely punishment for bad behaviour cannot be viewed as the only tool for stamping out such problems, but we still need to recognise that it is a tool and the message treating different groups differently for the same crimes sends.
On this last issue, it's also important to recognise indigenous laws and allow them to followed but not at the expense of our national laws and ethics. Barbaric punishments such as body mutilation should not be tolerated no matter how long they've been practised for. We need to recognise that our laws are based on what we think is universally right and if we changed our minds about what we think is right we'd change our laws. Especially on issues of serious physical harm, we should not compromise.
Affirmitative action is not really practised in Australia. I haven't studied it to any great length and I don't know a lot about the direct and indirect impacts. At the same time, we need to recognise that different socio-economic groups face different issues that require different responses. Viewing indigenous Australians as a demographic that needs urgent attention and having policies aimed directly at that group is definitely the most efficient and effective way of addressing the problem. Of course there are many other problems in Australia and addressing one problem does not to stop us from addressing others. While Australia is a noticeably more egalitarian society than the US, there are definitely still issues of poverty and social isolation. Some of the solutions to the issues facing indigenous Australians can be applied to the issues generally, some cannot. When reviewing those issues, both in indigenous Australians and non-indigenous Australians it becomes clear that those are the symptoms of deeper underlying problems. Addressing the symptoms doesn't always make the illness go away, and can result in a dependence to the drugs treating the symptoms (as we see in many communities). To solve these issues we can't just address the symptoms, we need to address the causes as well.