This is another question that I've been asked.
Why is it that American sports don't have promotion and relegation?
Because the baseball equivalent of the FA folded and rich owners stepped onto the scene who had an interest in keeping the national league as a closed circuit, admitting teams who would play ball with the other owners, and kicking out those (e.g. Louisville Sluggers, Cleveland Spiders, Washington Nationals, Baltimore Orioles) who wouldn't.
In 1903 another chance at establishing a tiered league system was firmly rebuked as the upstart (and then-superior) American League agreed to settle with the failing National League and collaborate as equals.
The big death-knells for any chance of a pro/rel system developing, though, were the 1922 decision Federal Baseball Club v. National League, which ruled that baseball, being an essential cultural institution, was not subject to antitrust regulations, meaning the owners' grip on the league and its players was absolute, and the 1926 World Series victory by the St. Louis Cardinals, which proclaimed to the rest of the league the effectiveness of a farm system composed of subordinate minor league teams. This victory kicked off a massive arms race throughout the rest of the 1920s, 30s, and 40s which saw the vast majority of independent and minor league teams get bought out and incorporated into existing major league farm system structures. With the death of the PCL in the 1950s, any hope of a change to the MLB franchise system was effectively dead for all time. The NFL and NBA developed within the context of the MLB's history, and so largely followed the same trajectory.
Consider what could have happened when in 1992 the biggest, richest, most successful clubs from the Football League broke off and formed their own independent league to protect their competitive and financial interests. Because soccer had at that point a 100-year tradition of promotion/relegation the format was ultimately preserved, but had that not been there, the owners that formed the Premier League would have had a financial interest in securing their position in the Premier League and thus would have pushed for a closed structure. That's essentially what happened to baseball in the 1880s and 90s.
tl;dr: the early histories of the soccer and baseball are remarkably similar, but four things happened to baseball that dramatically altered its trajectory compared to soccer:
1) The FA-equivalent for baseball, the NABBP wasn't able to handle the transition to professionalism and collapsed, with wealthy owners filling the vacuum left in the association's wake. This shifted the focus of the game from an open sport-wide system of various affiliated leagues of local, team-owned clubs, to a singular closed system of owners who had a deep interest in carefully controlling the format of their league (restricting players to teams, only admitting teams with large fanbases and amenable owners). Because the National League took in the biggest, most popular, most well funded teams, it was difficult for any other league to compete or negotiate, and there was no larger overarching governing body to arbitrate on their behalf, thus creating a Major-Minor league system.
2) Any attempt to break the existing format failed. Bad teams were tossed from the league. Rival leagues were either smashed in legal battle (The Federal League) or were incorporated into the structure (The American League), at which point they gained an immediate interest in maintaining the closed franchise system. Any attempt on the part of the players to organize and collectively bargain, was likewise crushed. Individual minor league teams achieving a level of play comparable to a Major League team (e.g. the Baltimore Orioles in the 1920s) were ignored entirely.
3) Federal Baseball Club v. National League codified the closed system into law. Baseball was a trust and could act as they pleased without having to entertain competition.
4) The success of the farm system established by Branch Rickey began a process by which nearly all minor league teams were bought out and forced to become subservient to the interests of the Major League.