Though I have a strong opinion on the issue, I think this Palestine statehood/ICC issue at best doesn't have a lot of relevance or at worst argues against your point because though the US objected, the measure still passed. This shows a lack of American/Western hegemony over the UN's operations.
I read a good Op-Ed on CNN.com this morning about Egypt's constitution and it made a point that is salient to this thread. The article sympathized with the Egyptian people over the fact that they haven't gotten what they wanted out of their revolution, which produced a constitution that would codify fundamentalist principals and restrict freeddom. The commentary was this:
The framers of the Egyptian constitution misunderstand democracy. Democracy is not about majority rule, it is about freedom, equality and the rule of law. While theoretically any system could provide these things, in practice none ever does except democracy. So that's why democracy exists. And the converse is also true: though Democracy is the only system that reliably provides these things, if done wrong it can restrict them just as surely as other systems do. And that's what happened (is trying to happeng) in Egypt.
That's what I see happening here, with the UN on Palestine and in other cases. The UN is doing democrcacy wrong. The UN is a democracy that does not protect the principles for which democracy was created to protect. Instead, it operates on majority rule even if the majority can't agree to protect those who need to be protected (Rwanda) or worse, acts or tries to act against them (Israel).
I'll liken this to a KKK rally, without intending an actual comparison of the KKK to another entity, besides simply the fact that it is unpopular. Rule of law, equality and freedom require a government to not only allow the KKK to demonstrate, but actively protect it from harm while it is does so. A too-literal interpretation of democracy would enable the police to either ignore a violent clash with counter-demonstrators (who typically outnumber KKK members in such circumstances) or worse, participate in the clash.
There are a lot more oppressive governments in the world than there are governments that protect democratic principles. And even some that profess to be democratic actually use democracy to suppress democratic principles. In my view, the ineptitude of the UN is largely a reflection of this fact. The UN does democracy wrong because most of the world does it wrong. But it is a two-pronged problem:
1. If the countries in the UN took seriously their mandate, then actions such as stopping the Rwanda genocide would be a no-brainer. They'd happen rapidly and with wide support, being driven by multiple parties simultaneously.
2. Since #1 doesn't happen, the US should do it and typically is the one to do it. People line-up and follow the big-kid, even while complaining that he's overbearing. The cruel irony though is that small actions such as Libya or Rwanda don't actually require the US to be the leader. Any major world power could have picked-up the UN flag and carried it into those fights*. For that reason, I consider criticism of the US by other westerners on that basis hypocritical, unfair and really just plain incorrect.
*Point of clarification: It is my understanding that we played a larger role in Libya than Obama wanted because our military has specialized operations capabilities others do not. But this just means:
1. Other countries would carry more risk if we weren't involved. France, who I undertand did want to lead the effort, could have done it, they just would have had to put their pilots in harms way to do it.
2. The lack of capabilities is a long-term implication of the same choice. It becomes a circular/self-reinforcing proposition: If a country like France chooses to step up and be a leader, they'll have to build their military back up in order to do it effectively.