Lots of reasons.
First, and primarily, there are a lot of sports, and sports teams, competing for spectators' attention in the U.S., and the subsequent ad dollars for TV airtime, stadium naming rights, on-field logos, etc. I realize there are other sports besides soccer in Europe, but the gap in overall popularity between association football and basketball or handball in Europe is much wider than the one between the big three team sports in the U.S.
There are also more teams in each top-tier sports league in the U.S. than in the English Premiere League (or anywhere else in UEFA), with 32 NFL teams, and 30 teams each in the NBA, NHL and MLB. More teams means more games means more airtime needed.
Second, and just as important, is risk/reward for the players. Gridiron football is a violent, dangerous sport, and to be picked in the NFL Draft says as much about your ability to avoid injury as it does for your skill. Despite this, ejections are relatively uncommon in gridiron football, so it's an exceptional case when a player is called to answer for causing injury to an opponent. The result is that, by the time a football player is drafted into the NFL, they may already be at the halfway point of their career in terms of the wear and tear on their bodies, which will stay with them the rest of their lives. There simply isn't enough reward inherent in playing the game at anything less then the premiere professional level to justify the risks the game presents.
Association football, in contrast, is technically a non-contact sport; aggressive actions made towards opposing players are penalized. As a result, injuries in soccer, though they do happen, tend to be more minor, and players who deliberately injure opponents are held to account for those actions more frequently. That allows soccer players to have careers spanning decades, rather than the average of about 10 years that you see for NFL players other than QBs and kickers. Now, this emphasis on injury being against the spirit of the game is seen in other U.S. sports, even hockey in more recent times, and you do see longer careers in those leagues, with basketball and baseball players continuing to dress for games into their 40s. However, the mentality of "move up or move aside" is still prevalent as players grow up through the school leagues; if you don't end up playing for the pros, there's not much money or opportunity continuing to play past your college years.
Which leads to the next reason; in the U.S., the player, not the team, is regarded as being mobile. The sports that do have notable minor leagues in the U.S., being baseball and hockey, typically have teams in various tiers that are either owned outright by one franchise or have partnership agreements. That's in addition to the prevalence of college sports teams in the U.S. (and by the by, if you want to see a loyal sports fan on the level of European soccer fans, don't look at the pros; look at college grads cheering their alma mater). The net result is that the player "grows up" through a series of teams, hopefully developing into a player that can compete in the premiere leagues, and the job of all the lower leagues is to help the player with that development, then let him go. In Europe, it's not just the player but the team that is mobile in the overall structure of the league. They player still grows, still develops, but the team that helped him do that is much more willing to try to hang on to him if they can, because this star striker or forward midfielder could be the ticket to a league championship and promotion.
We still haven't gotten exactly to why the U.S. hasn't bought into that same mentality, though, and the next reason is the one I think is most important; geographical coverage. The U.S, without Alaska (no professional sports teams in that state to speak of), is about the same size as all of non-Nordic Europe combined. That means that maintaining a national fan base for a sports league in the U.S. requires paying very careful attention to the geographic dispersal of your teams. In England, you'll notice the EPL teams are concentrated in the London metro and the Manchester/Merseyside area, with Birmingham growing up as a third center for English football. That's mainly happened through promotion and relegation; teams in urban centers have more spectators, get more ticket and ad revenues, can afford to hire better players, and become better teams, moving up into the premiere league and forcing out the teams from outlying areas.
Now, that's OK in a country with a total land size about the same as that of the U.S. Eastern Seaboard; if you want to cheer on a premiere-league team, there's one within a couple hours of pretty much any town in England. Or, as the commenter states, you can keep cheering for your hometown team even if they're not in the premiere league.
In the U.S., however, that kind of league management would result in a disproportionate number of teams being crammed into the East Coast cities, much as the EPL's teams have gravitated to London and Manchester/Merseyside. That would place a lot of large "heartland" cities, each with populations in the millions, at least a day or more away from the nearest professional team by car. With no minor leagues to speak of due to the risk-reward problems of many of these sports, support for the sport in any form in areas without a major team would simply die out, robbing the leagues of valuable TV time. That would be compounded by the concentration of teams in the most populous areas, which force teams to fight each other for the biggest slice of the revenue pie. As a result, teams are carefully managed by franchise owners and league leadership to ensure that the teams are spread as evenly as can be supported around the populous areas of the U.S., giving each team a built-in home crowd.
That, in turn, requires the teams to be kept on roughly equal footing; it's in the league's best interest to keep all of the teams it has financially healthy and competitive, and the two go hand in hand. So, instead of punishing franchises that can't buy a good team, we give the worst teams the first picks of new incoming talent. Instead of letting the richest teams self-perpetuate by spending more money on talent than anyone else can, we put salary caps in place to ensure no team can spend more than any other on their players. The current scheduling system even gives each team a game against a team from another conference they normally wouldn't play, with the same division ranking as their own, meaning the better teams have overall tougher opponents. This egalitarian approach lets dynasties rise and fall over a period of a few years each instead of a few decades (as is seen in baseball and in collegiate sports, where the school's reputation as a sports powerhouse is self-perpetuating over long timespans), and gives every franchise and every fan the best chance for their team's success.