Yes, companies will adapt to India's slower internet speeds. India has a huge population, and it is still growing. I think companies are aware of this, and they see huge profits in India. Companies will do whatever it takes to profit off of India's huge population, whether it means creating new websites or not.
India's population is huge. As more and more areas are able to have access to internet it is going to open a huge new population for companies to reach out to and make a profit. The internet is definitely an important part of daily life in the US so why not eventually in India.
Until India's internet speeds improve, companies wanting to reach and do business with people in the area have to work with the limits of the internet there. This has always been the case. Only recently has average internet speeds in the United States increased enough to allow full capabilities from many servers.
Although it's a good idea for your customers to be able to reach your page regardless of their internet speed, I cannot see this taking off as a worldwide trend. However, if LinkedIn Lite actually makes a big splash, you could see some companies in some sectors create a mirrored-type site that caters to those with slower speeds.