During the 16th century, The Jesuits had missionaries in the Chinese court. They were not successful in converting the emperor or his family.
What would have happened if they were successful?
Lets assume the pope doesn’t mind them venerating ancestors.
China with a moral objection to birth control? A lopsided Earth would wobble off its axis.
A more serious answer might be that China might not have suffered from Western exploitation to the extent it did. Nor would the “heathen” Japanese have been given free reign over a Christian nation in the early 1930’s. The ground would probably still be as fertile for 20th Century Communism/Athiesm as Russia was, though.
Ricci was around the court from 1601-1610 ( and apparently never met the emperor personally ), during the waning days of the Ming state. Any impression he would have made, which couldn’t have amounted to very much in just a few decades, would likely have been wiped out when the Ming went under to the Manchu Qing after 1644. The Qing, who were heavily involved in Mongol politics, on the one hand patronized Tibetan Buddhism due to those connections and on the other patronized court Confucianism to strengthen their legitimacy as traditional Chinese emperors. By contrast Catholicism or Islam offered little in the way of immediate utility.
Don’t fight the hypothetical. The OP simply asks what would have happened if the Wanli Emperor had found Jesus – not for political reasons, but as a sincere personal religious conversion. And let us further assume that he converted to mainstream Catholicism instead of coming up with some bizarrely confused version of the faith like Hong Xiuquan much later did.
If so, I suspect he would have felt obliged to discontinue, as pagan, the annual propitiatory rites at the Temple of Heaven . . . which was mainly what the Emperor was for, in Chinese culture . . . Therefore, he would not have lasted long as Emperor. Even autocracy has limits, and no Chinese Emperor (later than the First) had the power to play Constantine.
I’m not.
I’m saying if the emperor had converted it would probably have made minimal impact before the Qing swept into power a few decades later. And that the Qing would have had little reason to adopt the new court faith. Indeed traditional backlash by the Confucian intelligentsia may have prompted even more enthusiasm for the conquerors and perhaps a pogrom to extirpate this strange heresy that may have become associated with the Ming’s loss of the ‘Mandate of Heaven’. Or it would have prompted an even earlier fall to the likes of Li Zicheng or Wu Sangui.
I suspect if Christianity became institutionised, they’d eventually break with Rome and maybe set the Emperor as head of the church/Defender of the Faith a la England and the Anglicans.
They’d be unlikely to force out traditional Chinese rituals, they’d just plaster Jesus over it.
Would that be possible? The Emperor was, among other things, a priest, the chief priest of the state religion. (Which was neither Confucianism, nor Buddhism, nor Taoism; it was essentially an institutionalized form of the Chinese folk-religion predating all of those.)
Now, could the Emperor be a Christian, and still call himself the Son of Heaven? Seems that title should be reserved for Jesus. Nor do the above-described rites seem easily Christianizable.
Heh, it was far more likely that Japan would become Christian … it almost did. 
It’s praying towards heaven, this could easily be turned to praying towards god.
When I went to a megachurch in Hong Kong, they were praying for the congregation’s financial prosperity, which doesn’t seem too different from praying for good weather during harvest.
Christianity has adopted and adapted many pagan celebrations.
It’s not that bad, The Catholic Pope is called “Holy Father” and priests act as Jesus-subsitutes in certain rituals , so Son of Heaven doesn’t seem that bad for the Chinese pontifex maximus
When your going back 500 years and changing the facts, a whole different world emerges. Probably there would have been no communist revolution.
Probably no communism, period. That’s a pretty specific ideology that’s unlikely to appear in an altered version of human history that’s changed so far back.