About the 'what if': I doubt the Chinese would have gone for Europe. If we think what the Europeans did, they went straight for the wealth (China and India, by mistaken way of America). The Chinese may well have gone for the East African coast (Zanzibar, Mombasa) and in particular - dominance of the Arabian Sea. This is where the world's powerhouses (and hence naval booty) was between say 800 - 1500AD. Naturally, as Europe did, they may well have churned the revenues from these victories inward and strengthened domination of the Silk Road. Europe was neither a threat, much of a prize, nor was it actually practical at that time to go and 'take out Europe'. I doubt they would have gone there.
As for crossing the Pacific. They knew the world was round already, so America would have been hit for sure. I'd probably anticipate the Chinese 'Conquistadors' doing a similar, if not more ruthless job of the Native Americans to be frank.
Gross hypothesis but it is a 'what if' thread

. Oh and this all of course takes the European naval supremacy model, rather than thinking particularly of Chinese interests at that time. I don't really know those well enough.