The Ming Dynasty had, reportedly, one of the largest navies in the world during the 15th and 16th centuries, which is when the "New World" was being "discovered" by Europeans. They also had made contact with European traders during this time, and had surely heard about the Americas that were being visited by the Spanish, English, French, etc.
Why did they never send any fleets across the Pacific and try to claim a piece of the "new" landmass that was being explored during this time?
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They didn't try because it wasn't politically relevant to them (i.e. The Emperor wasn't interested). Chinese dynasties preferred a tributary network instead of European or Pan-Arabic style colonisation.
This reasoning worked well enough considering the key motivation for Europeans traders to sail beyond Europe was to bypass Arabic tariffs on the Silk Road and connect with Sinae (China) directly. From the Emperor's standpoint, foreign traders came to them.
The Chinese tributary system didn't collapse until the 19th century Opium Wars pushed European foreign policies on China.
Tianxia ("All is equal under heaven") is an intriguing political concept and explains much of China's diplomatic philosophy to this day; and at the risk of drifting into History's twin, Futurology, is proving appealing to certain emerging nations that are more comfortable with Chinese soft power instead of American hard power.
It's a much more complicated process than saying "Let's do it.", to provision a sea voyage of (at that time) likely 12+ months, return trip.
Accurate maps didn't exist at the requisite scale, and pilots "chart's" were patentable; so a European pilot (or three) would have been advised.
