A global government isn't really possible. I think the fundamental issue is that a tribe of "everyone" doesn't really work without a counterpart. I think the solution begins by colonizing Mars, a few moons, maybe some asteroids.
I think a better way to frame this is “Is it possible to use the rule of law across national boundaries?” Clearly the answer is currently a qualified “yes”: Laws, treaties, etc do exist and are commonly used. The areas that are addressed are clearly not uniform, nor can we rely on all nations to participate, and the enforcement of laws across national boundaries are extremely tricky and currently limited. However, that should not stop us as a planet from trying to improve global cooperation through the law, rather we should look at it as “more work to do”.
Global government is only the extension of local, national and regional government. The E.U. already is a kind of "international" government in that it creates de facto laws, rules and regulations that supersede the laws of its member states. Similar constructs (though not as advanced) exist i.e. in West and East Africa.
A global government is an entirely logical next step and could be a very valuable asset when dealing with truly global issues.
If I'm living under a dictatorship, at least I can try to escape and move to a better place. If a global government becomes tyrannical, where do we go?
Such an idea is centuries away in the best case scenario.
How about you stay and work towards changing the government? This is literally how every democracy has developed. It is also the reality for several Billion people today. Most can't just up and leave if they disagree with their governments. Borders are not open for most people.
I'm far from convinced that a global government could possibly be a good thing. I think that a large part of the political problems in the US, for instance, is because its trying to govern too many people of very different and often incompatible cultures and values.
The magic word is "subsidiarity": the principle that political decisions should always be made on the most local level that still enables their resolution. Under that principle, a (democratically legitimized) world government would only be tasked with creating laws pertaining to truly global issues (i.e. setting limits for the emissions of CO2). I agree with other comments here that this is unrealistic in the near future. But that doesn't mean that it is not a good idea.
The trick is coming up with a system where the upper layers of government don't try to take over the matters from the lower ones. The problem, fundamentally, is deciding what "pertains to truly global issues" - in a democracy, it's ultimately the people doing that, so if you can convince them that whatever local problem X is really important, it can become "global" all of a sudden.
This kind of stuff is usually set out in constitutions or basic laws which in turn can be enforced by courts – the usual division of power stuff. You could also introduce some sort of check on the power of a global government. In Germany, parliament has to get approval of the majority of state governments for certain laws to enter into force, for example. Same is true for the E.U.: The E.U. parliament and the council of the national governments of E.U. member states both have to agree to new laws and regulations in most cases, which guarantees that either one can't ride roughshod over the core interests of the other.
It usually is, but long-term there's a trend of, shall we say, creatively reinterpreting what's written to the same effect - just look at US. Some would argue that EU shows the same trend, although it's young enough that this doesn't manifest quite so much.
I’d actually argue that the increasing centralization of power in the U.S. is due to the increasing nationalization of political issues and the requirement for nation-wide resolutions to problems. The way it has expressed itself (strong presidential executive instead of parliamentary democracy) is probably due to the way the U.S. constitution was originally framed and some random events asking the way.
But that's the thing - why are these all suddenly "national issues"? Even healthcare can be done state by state (Canada of all places did it that way, and their system is still fundamentally province-centric), never mind all the culture war stuff. Is it really the most local level on which these matters need to be resolved? Or are they deliberately pushed there for political games?
Edit: fine, more Mars land for me!