It’s difficult to support the claim that the American rail network is "abandoned." It’s almost as large as that of the entirety of Europe, though we have shed (as redundant or unprofitable) nearly half the trackage we once had. It’s just that the US has chosen to keep its network under private ownership, and in recent decades has not chosen to subsidize a new passenger rail network.
Railways originated as private enterprises in the United Kingdom and, shortly thereafter, in the US. As infrastructure undertakings of a scale not previously known, they were among the first enterprises to need a new legal concept: that of the private corporation. The industry, like many breathtaking new inventions, attracted a lot of “irrational exuberance” as well as outright fraud in the organization of the business enterprises and the construction of the lines. Soon, legislators reacted with securities regulation, and many states also forbade public subsidy to railroad companies who—like tech companies today—would pit towns against each other to see which would subsidize most highly a route through their city.
In Europe (and India and Japan), the decades of railway building coincided with the 19th century period of creating modern nation-states, and so railways in financial trouble were more readily brought under state or royal control. In some European nations this happened almost immediately, while in others it occurred in the wake of World War I. The US (and Canada) also built a number of railways to tie together the nations, notably the first Transcontinental Railroad (completed in 1869), but primarily did so by offering grants of free land rather than direct subsidy or taking ownership. Even though American railroads were receiving all kinds of government aid, US politics and public opinion steadfastly treated them as private enterprises, and a brief period of government control during the First World War was always seen an a temporary exigency to be ended as soon as possible.
Geography also played a role. Europe, India, and Japan are places with great density of settlements, meaning not only megacities could easily be connected by rail, but hundreds of smaller cities as well. Only a few regions of the US have similar settlement patterns. On the other hand, North America has vast seas of emptiness to transport coal, grain, petroleum, chemicals, and finished goods across at low cost. Unprofitable passenger operations were abandoned as soon as regulators would permit, with a few surviving routes passed to Amtrak in 1971. In this sense, the US is not terribly different from the nations of South America, South Africa, or Australia. In the US, about 35% of freight moves by rail; in the EU only about 10% does. Thus, North American railroading came to serve the needs of freight shippers, and of railroad shareholders. In the last decade, new pressure from shareholders has pushed US railroads to emphasize moving lengthy trains with a minimum of labor inputs from terminal to terminal, even closing their classification yards. That means we've abandoned to trucking not just less-than-carload freight shippers (that was mostly done by 1970) but nowadays even those shippers who send out a few carloads each month have trouble getting railroads to return their calls.
For complex interconnected reasons of politics and national pride, a few nations have in the last 50 years chosen to make significant government investment in passenger service, usually entirely new networks of high-speed rail lines. This era began with Japan’s 1966 opening of the Tokyo-Osaka Shinkansen; gained new impetus with 1980s/1990s decisions in France, Spain, and Germany; and spurred smaller networks in Scandinavia, Italy, Southeast Asia, South Korea, and even North Africa. All now have been completely dwarfed by the construction of an enormous new network in China. Political calculations in the US, however, have so far not led to such investment in a national high-speed network. From the postwar era until recently, intercity highways in the US—though not all local roads and streets—were financed entirely by highway users, making a huge subsidy for passenger trains difficult to defend. Energy and environmental considerations that arose in the 1970s just happened to be followed immediately by deregulation of airlines in the US (1978), which led to much intercity demand being met by airlines that received little visible subsidy. Additionally, American land-use patterns don’t support city-center to city-center trips by rail very well. All this reduces the number of city-pairs in the US (and many other nations) where a robust intercity rail market would exist, with short trips bled off by private autos and inexpensive bus service, and very long trips only really practical by air.