The idea of a "developing nation" is a really modern term and one that I would probably associate with the legacy of Cold War politics (but I'm not a 20th century historian so I might be way off there). But I gather that you're probably asking more about when the US came to be a larger, more powerful economic force in the world which could compete with other larger powers.

That shift for the US takes place over the 19th century. Around the time of the Civil War the US is one of the largest world economies, behind Britain and Germany specifically. By the end of the Second Industrial Revolution a couple decades later the US has an economy several times larger than Germany and Britain combined. The US very much grows by leaps and bounds during the late 19th century.

But I would say that the US makes the shift away from a minor "developing" economy even earlier than that, during the 1820s and 1830s in what historians call the Market Revolution. This period is when the US feels the most effects of the first Industrial Revolution and transforms itself into a fully capitalist society, by which I mean market forces become the driving force behind production, prices, and economic growth.

The Market Revolution is made possible by advances in transportation and communication technologies like the telegraph, canals, cotton gin, and later railroads which allow Americans to transcend local economies and embrace larger markets. Increased manufacturing and transportation ability means that certain goods can be made outside the home and purchased through consumption. Crops can be transported long distances without spoiling and prices can be monitored over long distances, meaning farmers can concentrate on growing more profitable commodity crops and use the proceeds to buy other food and goods that they need.

In short, the Market Revolution allows the US to better integrate its economy on a regional or national scale, as opposed to local relations. This is also the era when economic activity and output in the US begins to take off and become relevant (and eventually competitive) in the eyes of European nations. These two developments are certainly related.