I'm pretty sure Chinese ships would be at least as capable as European ships of crossing the Pacific. As far as I know, Zheng He's fleets did not always stay in sight of land - they crossed parts of the Indian Ocean on the Monsoon winds. Even if the size of some of Zheng He's ships has been exaggerated, many of them were probably as large or larger than the Spanish galleons that started to cross the Pacific in the 16th century, and a lot larger than the ships that Magellan first crossed the Pacific in.
The real problem, as others have noted, is motivation. Zheng He's voyages were made to lands that were well known either to Chinese merchants or to their immediate trading partners, so they had a good idea of where they were heading and to a certain extent what to expect when they got there. The Americas (or Australia) would be a different story. There would be no obvious commercial motive. China had no historical tradition of establishing overseas colonies for prestige reasons, as opposed to establishing diplomatic/tributary relations with already known foreign states. The relatively tolerant religious climate would make it very unlikely that there would be a religious group that would find the situation so hostile in China that they would be willing to consider moving to a distant land, even if it were already known about. I don't know if China had any tradition of setting up something like penal colonies, but if they did, there were probably plenty of pretty remote places within the main Chinese Empire to send the convicts (Mongolian frontier, northern Manchuria, the far west, the edge of Tibet, ...) without having to consider sending them thousands of miles overseas. The Chinese also never sent out expeditions for pure discovery purposes - just to find out what's out there - but neither did the Europeans, at least until the late 18th century and the time of Cook and Bougainville, and even those expeditions were not devoid commercial motivation.
Two possibilities occurred to me. 1. The Chinese, at some earlier point in their history (perhaps the Song dynasty), manage to learn about parts of the American continents by accident (unlikely, but possible - a ship or ships would have to get blown way off course, get pushed eastward all the way to North America, and then persist in trying to find a way back home, avoid shipwreck or other catastrophe, find the tropical currents going westward at a lower latitude, ride them all the way back to Asia, get to China, and manage to get their account widely known, with influential people taking notice). Sporadic contact is maintained for a while, then lost. When the Ming decide to send out naval expeditions under Zheng He or an equivalent figure, they first cover ground in southeast Asia and around the Indian Ocean, but after they have done several expeditions to that area, they decide to send an expedition on a more daring trip to reestablish contact with the distant lands to the east and their strange peoples that are still well-documented.
A very different possibility, which shows up, for example, in Tony Jones'
Monarchy World timeline, is to have China become interested in colonization much later, after the European powers have already taken much of the Americas. A series of reforming Emperors in the late 17th or 18th century decide that China can learn things from outsiders, and one of those things is that overseas colonies can be very good for trade and the economy. The western coast of north America in the 18th century could be ripe for setting up a Chinese colony - the Spanish aren't in California in any numbers until mid-century, and even after they come a strong Chinese expedition could probably overcome resistance. If they don't want that conflict, everything from northern California to Alaska is virtually untouched by Europeans, apart from vague claims.