No but not for the reasons given so far.
The main - and major - reason that it won't take off is geographical, plain and simple. In England, you can have a lower level league include promotion and relegation in the main because every single team in the division is CLOSE to each other. For instance, how would you fund a grassroots side playing one day in Seattle then the next in Baltimore, 2770 miles away in a national league in the United States? You simply couldn't. The worst we have in England is maybe Plymoth travelling up to Morecambe or something, and even then that's pretty horrendous an undertaking - but it's still 'only' 330 miles or so.
Realistically having a national league whereby every team plays each other nationwide on such a level is not possible. The only feasible way I can imagine is state-level football with their teams including promotion and relegation, within their own separate jurisdictions. California, Florida for instance could definitely have a decent enough league structure of their own, and then eventually form their own United-States style Champions league. But why bother with all that when you can skip all that and jump STRAIGHT to Champions League style games, with a playoff system instead of bothering with the endless struggle it would take to forge natural pyramids in each state?
I could be wrong and the grassroots game at state level in the United States could be stronger than I think, but (without living there obviously) my assumption is that generally people in the USA have a stronger national identity than state identity and - I suspect - would prefer to rally around one or two 'champions' of their state rather than watch lower level sides from their towns and cities fight it out.