Looking at countries with no minimum wage, or, to put it another way, no formal organisation(s) to lobby for the rights of workers, it is clear that the results are less than savory (hint: the only country this applies to is North Korea) - http://en.wikipedia.org/wiki/List_of_minimum_wages_by_countr...
I think that the core issue is that the majority of people expand as much as they can until something stops them, which is why formal agreements are necessary at this stage of humanity's development. Few people are conscious enough to act fairly in the absence of law. I think that the long term solution to creating a society where a minimum wage is not necessary is education and a culture of personal development. When people stop outsourcing their morals to authority, they will make the right decisions.
Funny thing about comparing the US and Sweden, the US has a minimum wage and Sweden doesn't. What Sweden does have lots of, though is a lot of re-distributive taxation[1]. If you go and look at actual real world economists instead of the caricatures that make it into our political debates in the US you'll find that most of them are actually democrats, and many like progressive income taxes, welfare, etc. But despite the fact that most of them like government spending, pretty much none of them like the minimum wage. You could say that Sweden is exactly how your typical democrat economist would run a country if they could, with Singapore being the conservative equivalent.
[1] Ok, their taxation structure is less progressive than the US's, but they spend much more of their taxes on services instead of, say, fighter planes so in practice its more re-distributive.