The Constitution replaced the inadequate Articles of Confederation just as the UN replaced the inadequate League of Nations. The Constitution created a continental nation, the Charter did NOT create a global one.The dirty little secret is that the Constitution of the United States is the best model of a union of nations. Given the unpleasantness of the early 1860s, of course, the issue is: better than what?
If you were serious about growing a world government, the first thing you would do would be to resore the Constitution's design by repealing the Seventeenth Amendment. This would stop and ultimately reverse the subversion of State's rights which has proceeded apace ever since the senators stopped representing their States and started representing the people living in their states. Others would call other Amendments into question, with reason--but I think repeal of the Seventeenth Amendment is the minimum.
From that basis it would then be practical to speak of recruiting additional States, some of them outside the Western Hemisphere, into the Union. First candidates would be the members of the British Commonwealth; Canada, Australia, and Britain itself. There are of course details such as the monarchy to be handled, tho if the Constitutional restrictions be rightly understood to apply in nearly all cases strictly to the Federal Government itself--and not to the States--there would be much more room within the structure than now seems to be the case.
The practicalities would seem to dictate that large nations (e.g. Canada) could, before joining the Union, first break themselves down into their individual provinces to have reasonably proportionate representation in the Senate. But that would call into question Statehood's actual purpose. Already, California is comparable in population to a great many nations, and in economic terms to all but a few.
While I'm indulging in fantasy, I'd better propose that Governors of States be made into e-Senators. Certainly we need to have some reform to show for the wrongful acquittal of x42! Would we really want union with a country willing to be led by such? Governors are far more nearly peers to the president than senators, certainly than popularly elected ones. They would, we could hope, be unwilling to vote to acquit for fear of the logical implication that their constituents could expect no better character in themselves than x42 in fact exhibited.