When word spread of the discovery of the New World, why didn't any Asian nations try to colonize it?

The way that you have framed your question suggests to me that you are looking at it with a modern perspective. In this instance it's limiting you from understanding the answer to your question.

The key thing to understand about this time period is that everything was motivated by a desire for trade. The voyages of discovery were conducted by the Western Europeans to try and find better alternative routes to India, in order to cut out the Islamic middlemen. Not to settle new lands - that was a byproduct of their actions.

The opinion that 'colonies = good' is anachronistic, a modern trapping that wasn't true of that time period. The Spanish wanted to plunder the native Empires and extract it's mineral wealth. The French colony in Quebec was nothing more than a trading post. The English Virginia settlements were established with the intention of finding another wealthy empire and taking their gold, like their Spanish rivals did to the Aztecs. If the colony didn't achieve it's intended purpose, it was often neglected and sometimes left to die.

Why didn't they go on their own voyages of discovery?

The Chinese did, just earlier and in a different direction. Zheng He is one of the most famous Chinese explorers. He led the vast Treasure Fleet as it explored all the way to the east coast of Africa, charting the lands and exacting tribute from those that he encountered. The Treasure Fleet was later scuttled by another emperor who felt that there was simply no reason to explore.

China (Zhongguo) literally means 'center state', a name that reflected their position in the Age of Discovery. Ming China was the dominant power of the world at this time and trade sought them out, not the other way around. The Chinese state was content with it's existing demesne and it was a common perception that the rest of the world offered nothing to China that it did not already have.