As the analysts are finding the last couple of years, India and United States are much more common than what most people think (did I just say that the world's poorest & richest country have some commonality). Superficially, India's english, democratic tradition & British colonial legacy are visible. But, we have just got started. There are much more deeper. We will randomly see them and then put them in place.
1. Hollywood Vs. Bollywood - You must be thinking that I'm just kidding, as movies are not a factor for world economy. It is. At one point, not long ago, Russia overtook US as a super power and Japan overtook US as an economic power. But, how many kids dreamed of Russia & Japan, tried to imitate their culture, flock to their universities or even learn traces of Russian & Japanese? But they did all that to US, because of a very powerful influence - Hollywood. It shaped the world cinema, and gave a way for imagination and thought. In all that imagination and dream world, only America was there and its perspective on the rest of the world. Interestingly, whereever they lived most kids saw world from the prism of United States. Thus, US got the best of world relationships, best of students & scientists & entrepreneurs and easily outsmarted its competitors. I'm not saying that Hollywood was solely responsible for America's growth, but without hollywood, a lot of American succcesses would have never happened. For a long time, US benefited from the export of Hollywoodism, McDonaldism & CocaColaism. Culture shapes billions of bucks and US companies & brands (from Ford, GE to Pepsi, Coca-Cola to Microsoft) became a marque for the world.
Coming to India: India's long lasting legacy in Asia it is its export of culture. Everwhere from Thailand, Malaysia to China, Japan, Middle East, coastal Africa... you could see great influence of indian culture. Thus, even without magnificent war & army India had conquered a huge territory, just by being a powerful holder of cultural exports. People from Columbus, Vasco Da Gama to English sailors to Chinese buddists flocked to India and dreamed on a trade with India. In a way, it is similiar to American export of culture, only a good form of culture :). And as Bollywood emerges, people from as far as Africa to East Asia will see world from an Indian prism and will bring both development and international relationships with that, and Indian companies will get the visibility and brand promotion it requires. Probably we might even convince Pakistan and China to have better friendhship with India.
This is also extendible to other media. Washington Post, Time magazine & CNN has so much effect on world media. Now India has many of the strengths and can take a part of the strength just how US took from UK (with BBC, Times etc.).
2. Enthusiasm & Entrepreneurism: If a Rip Van Winkle from 19th century America suddenly woke up in current India, he could find the comfort of home in it. 19th Century America didnt have infrastructure, had shabby roads and dusty towns, quarelling federal government, but its people had an unique power: An optimism for the future, a sense of pride and vision & an overbearing urge to succeed. They quickly moved from the heterogeneous group of colonies to bring powerful economic houses that dreamed of growing big and conquering the world. This fiery capitalism and energetic entrepreneuship is what we see in India - from slums to high raises. We are poor in infrastructure and shabby in our cities, and our heterogeneous is quarallesome, but our energetic young men & women are going to overcome all of that with the same sense of purpose our godfathers (19th century Americans) had.
3. Stress on Unity in Diversity: America has a great power, it quickly assimilates its immigrants much faster than any other country. The irish, jews, Italians, Germans who came during the last 100 years are now fully integrated as Americans. Though, there is great diversity in terms of immigrants (we have huge African-Americans, Hispanic-Americans, Asian-Americans and of course European Americans) representing every continent in the world. But, they all are almost united on a concept: American (albeit diversity and fractionsim are growing).
Historically, only one other country had such a track record: India. It had such a power that whoever came to the land became Indians (or Hindus). We definitely have an exceptions like Islamics who will never be assimiliated anywhere. The Greeks, Afgans, Aryans, East Asians, Mongols, Arabs are all fully assimilated and the art & culture as an example for it. Thus, it has such a diversity (in language, religion, art forms) but a soft thread runs among all, conscious or sub-conscious. A sense of Indianism. A lot of Indians will oppose when I say it, till they feel their subconscious emotion of it at different times.