I work on the history of Anglo-American relations and this question (or at least a variation on it) interests me greatly. At what point was the United States' rise to power perceived to be inevitable?
Firstly, it all depends on what you mean by "set up". On the one hand, you could argue that America's ability to compete economically against its European rivals owed a lot to its natural resources and capacity for growth. In this respect, the country started life with a number of geographic advantages. If the United States had been founded in a smaller area, with no room for Westward expansion, then it probably wouldn't have been able to challenge other world powers in the same way. That said, this kind of geographical determinism only goes so far - after all, the key is how a country uses its resources.
In the immediate aftermath of the Revolution, some Americans were already beginning to make wildly optimistic predictions about their country's future. There was lots of talk of sweeping away the crumbling, decadent 'old world' of Europe and replacing it with a pure, vibrant, republican model of society nurtured in the new world. However, for much of the next century these comments were laughed off in Britain as the exaggerated boasts of loud-mouthed Yankees.
From a British perspective, the 'inevitability' of America's rise only started to become apparent from around the 1870s. At least, that's what I argue in my own work. Before that, the country was generally viewed as an underdeveloped frontier territory and an unstable experiment in republican democracy. British observers watched the country with great interest - and many, of course, went to live there - but it's hard to detect any sense of anxiety about its future power. With hindsight its easy to laugh at their confidence, but we shouldn't forget that post-Revolutionary America was an extremely unstable country and that it came close to tearing itself apart during the Civil War.
However, after the Civil War things began to change. By the 1870s people in Britain (including Gladstone, the Prime Minister) were already starting to talk in terms of a coming 'American Future' in which the United States would inevitably usurp Britain's position on the World Stage. These anxieties manifested themselves across a range of contexts (economic, political, cultural) until, by 1902, the 'Americanization of the world' was casually described by one British observer as 'the trend of the twentieth century'.
America surpassed Britain in terms of industrial production in the 1870s and started to eat into its overseas markets. Hundreds of rich Americans were marrying off their daughters to bankrupt families of the British aristocracy. In cultural terms, British audiences were beginning to demonstrate a preference for American pop culture (particularly its humour and its journalism). By the 1880s, performing cowboys like Buffalo Bill and Mexican Joe had captured the imaginations of millions of Victorian girls and boys who now started to dream about life on the American frontier. Visitors to the Chicago Worlds Fair of 1893 returned with stories of an electrifying modern metropolis whose streetcars, elevators, and skyscrapers made it seem like the landscape of the future. American sportsmen were also starting to get the better of their British competitors in yacht races and boxing matches.
Another indicator of America's growing influence in this period was the spread of American English. It might be hard to imagine, but hundreds of American phrases and slang terms entered British culture during this time (including my username). For more on this, check out an article I wrote for The Guardian a while back.
All of these manifestations of modern American vigour fed into growing concerns about the degeneration of 'old world' society and culture. Whilst many people in Britain resisted the idea of an American future - it would be wrong to suggest that the whole country was resigned to defeat - by the 1880s it was in wide circulation. The obvious analogy would be the recent emergence of China - America is still the world's top super power, but people are starting to talk openly about a power shift towards Beijing.
So, to answer your question: while some of the raw ingredients of America's future power were in place before the revolution, it wasn't until the final quarter of the nineteenth century that its rise to global domination began to be regarded as inevitable by the international community. Events such as the First World War helped to accelerate this process, but it was already well under way by the end of the century.
If you'd like to know more about any of this, take a look at:
My PhD thesis on transatlantic journalism - free to download!
This article tracking the transatlantic circulation of American jokes - free to download for a short time.
Various other bits and bobs on my research blog.
Some previous /r/AskHistorians threads on these topics (I've copied some of my responses into this post already):