What, as in... doing so tomorrow?
Or in the distant future?
Given the time to develop propulsion technology, what is so LOL about extra-solar colonization?
Propulsion limitations are the only thing technically holding us back. That's a pretty big wall that will probably require a few centuries to scale, but once we can attain fast enough travel, the first stages will be like ancient sailing missions across the ocean, requiring weeks if not months.
And, last time I checked, even with ancient infrastructure, we found a way to support such colonies.
Baby steps.
If we can get to that point, we'll already have inner-space quite well conquered, and will have orbital colonies as well as lunar and martian colonies established.
If that stage of our civilization arrives before an "on the brink" style of WWIII, and occurs before an earth-destroying threat (at the hands of nature or aliens), then space habitation will drive us to unite under a single banner.
It may not be entirely voluntary and bring great amounts of glee, but unless it's a militaristic-regime performing a coup, I think it'll be at least hesitantly welcomed. As mentioned, trade and fair practices will mandate such an oversight.
And as orbital colonization and planetary colonization expands, the threat of conflict will rise yet again. The curious question is, how long after that does a new/evolved united government come into its own? As in, so the Earth government and the non-Earth government(s) didn't agree and had a spat - does it take a few times trading blows before a United Mankind government forms, or does that even happen before it can escalate?
What if there's enough intelligence to create a top-level government from the moment the colonies have grown enough, one that overseas everyone but then let's the Earth and Colonies create their hierarchical structures and manage their affairs until it's an inter-party affair.
The analog of the original U.S. experiment is very much what I see as the overall format for a global governing body, perhaps with a few twists. It'd definitely need to be a federal republic, especially in the early stages so that cultures and governments could maintain some autonomy and remain unique.
The cultural transformation of an evolving U.S. would be appropriate as well, so as to minimize conflict due to stagnant cultures and religious differences, among other causes for concern. The thing that would not be welcome in the "evolving U.S." model, is the political infection that followed. The world body would need to be careful about crafting a representative/electoral system that minimizes both the odds and impacts of such an outcome. It's a little daydreamy, I realize, but I think not impossible given a rigorous crafting process and care to prevent any loopholes that allow for great change.
You could almost create an "unalterable" government. Allow for new laws that fit very carefully into an extremely rigid web, but don't allow people sweeping changes, going go far as to remove certain democratic government approaches (citizens voting for such sweeping changes).
I think a carefully crafted republic, with less democracy, has the potential to be a favorable global governing body. Leave the democracy flourishes for the lesser governments, but the top-level needs to be a well groomed, carefully crafted and very limited republic-style empire. We know the history of large republics falling in on themselves, but they grew into that position.
Make them a certain static size, with a certain level of expansion written into the books, with no room for sweeping changes and growth in power... unless civilization itself falls on its face, that "empire" shouldn't face the typical empire struggles where they too big or too powerful.
It would be interesting if we/they could also write into law one, and only one type of sweeping, radical change: a mandated vote every so often (once every three decades, perhaps?), or maybe a petition to put this up for vote for all citizens type of event... a vote that says, should a committee be drafted and a large army drawn, in order to create and protect a new era for the republic, to be introduced into regular law 10-15 years after this date.
Require a massive majority (4/5's ?), and... do something fancy with laws and policing to ensure transitions are smooth. Keep a certain structure always the same but allow those committees to change certain things, expand this or reshape that, etc...
Just kind of flowing new thoughts to text here... haven't given it much thought. Seems an interesting thought... this thread is fairly pointless as is, might as well make a thought-exercise out of it.