Can they? Tibet is the most peaceful place in China, what can Indians do? Radical Muslims? they hurt India much much worse than they ever did to China. Xinjiang is much safer than anywhere in India.
And you think promoting separatism in India wouldn't harm China in the long term? These things almost always backfire badly.
Think about it.
Your country has at least 11 million Uighurs, although some organizations say that there are at least 15 million Uighurs in China. Almost all of them live in one of the most remote and rugged regions of China. To make matters worse, the birthrate of the Uighurs is much higher than that of the Han Chinese. Han China is currently facing an aging population and manpower shortages. The Chinese government is desperately looking for ways to replace workers with robots or encourage overseas Chinese people to return to their country. It's also beginning to attract Han Chinese settlers from remote parts of China (e.g. Xinjiang) to move back to the coastal eastern regions in order to compensate the manpower shortages and fill up the many ghost cities.
China has a lot of problems, but the censorship in your country plays a huge role in giving outsiders the impression that everything's fine and dandy.
You also have to deal with, potentially, thousands upon thousands of radical Uighur jihadists who are currently living in Syria along with their wives and breeding like rabbits. Unlike India, which is geographically protected by huge mountain ranges and has direct access to the open seas, China is geographically protected by desert regions to the west and choked by a chain of islands to its east and southeast. That explains why China is trying everything in its power to break through the South China Sea, since it knows very well that it can be choked by hostile countries such as Japan if those island chains don't fall under Beijing's direct control.
Why the heck would China risk starting problems with India while it knows that the US/West and Japan are looking for practically any excuse to go against (and isolate) Beijing?
The separatism situation in China is much, much worse than it is in India. Tibet is only peaceful right now because India and the US haven't decided to destabilize it yet. And Xinjiang can very easily turn into another Central Asian jihadist hellhole.
It would be very unwise for China to play with fire.
Instead, China and India should become more cooperative and ensure that the 21st century becomes the Asian century. Promoting separatists in neighboring countries will backfire on China very quickly.
India has enough manpower to fight long wars for the next 50 years due to its rapidly growing population. China's era of rapid population growth is over. Your country can't afford perpetual warfare.
A little bit of wisdom and humility would be nice.