To begin with, it doesn't really address the real reason for America's ascent and descent. Briefly: we have, by far, the world's largest economy. We have the world's third-largest population. Given those two factors alone, you'd expect us to be in the global driver's seat, and neither of them is going to change anytime soon.
Even with the rise of India and China, does it make any sense at all to think that a country with a GDP the size of Japan+China+Germany+France is going to cease being highly, highly relevant? That a country with 300 million relatively well-educated, prosperous consumers is going to stop driving global demand? That such a country, which boasts (for the well-off) one of the highest standards of living in the world, along with some of the most desirable living spaces on the planet, will stop attracting foreign tourists, immigrants and job-seekers?
The foundation of most of this "India/China is the new America" worrying somehow assumes that, in the future, people will stop wanting to live in New York and San Francisco and Los Angeles and Chicago, and instead want to live in Mumbai and Bangalore and Beijing and Shenzhen. For people who grow up in India and China, that's a reasonable assumption, but having visited one of those countries (and supposedly the easier one for an English-speaker to get around in) I can tell you it's still a pretty huge leap to make for a non-native.
So why does it feel like "America's moment" is passing? Well, because it is. Our days of being the only dominant economic power in the world are coming close to their end - but is that a bad thing? Much of China and India remain desperately poor places, where human suffering and misery is vast and at a level nearly unimaginable in most of the United States. Safe water and sanitary living conditions are far from the norm for millions and millions of people in these countries. So wouldn't it be a good thing if they built more companies that started raking in some of that juicy foreign currency that could help them provide basic services to the poorest among them?
And as China and India become more middle-class and moneyed, isn't that a good thing from an American perspective? Millions and millions more consumers for our companies, our culture and our values. Millions fewer poor, unemployed young men for whom radicalism appears to offer the only way out of grinding poverty. Eventually, a higher standard of living that puts an end to the sweatshops and Dickensian factories, and reduces the salary gap that makes it so attractive to outsource American jobs.
But more than anything else, the rise of the Indian and Chinese economies simply means less suffering for millions of people, and frankly, thank God for that.