I think it's due to no notable rulers in the east converting to Christianity and allowing/supporting missionary efforts.
You make it sound as if Christianity had become the majority religion in three hundred years in Rome. It wasn't, instead what happened was Constantine became a Christian and gave state support to the Church. Christianity had grown and become more influential but the conversion of an emperor and the promulgation of Christianity by him, his sons (though they were Arian) and Theodosius made the Christian tide in the west nearly irreversible. Same can be said of the after the fall of the western Roman Empire. Christianity was established under the Franks through the conversion of Clovis. This also goes for the conversions of the Rus, the Scandinavians, the English and others.
If an emperor of China had converted to Christianity and managed to have a significant grasp on authority you might have had a Nestorian Christian China. Same goes for the great Khans. Heck, if the Taiping rebellion had succeeded or Christianity had not been utterly banned from Japan there might have been a Christian dominated Asia today.
There also be something to the more developed religions of the east than pagan religions in the west with their vague mythologies of Zeus or Thor.