Why aren't Christians still debating the nature of Christ, and why did this theological debate not cause schisms in the Western church like it did in the Byzantine Empire?
One answer is that there wasn't really a rival strong enough to challenge papal interpretation. The only other sees which ever really tried were Ravenna, the seat of the Imperial Exarch, and Milan, working off the legacy of St. Ambrose inter alia. These really didn't have patriarchal standing, so they were not truly significant challenges, and in any case they were by and large political, rather than belief-based disputes.
Another answer is that the papacy prided itself - and still does pride itself, even Pope Francis (no, 20 year rule, don't hurt me) - on changeless adherence to orthodox doctrine. This is not really true, but it does mean that fairly obvious theological innovations, like Monophysitism, were never going to get papal approval.
Yet another answer is that debates over the nature of Christ are still very much alive in Christian discourse today. For example, Lutherans believe that Christ was half human and half divine, while Catholic doctrine holds that Christ was fully human and fully divine. No, it's not supposed to be comprehensible to the human mind.
Another (!) answer is that Monophysitism was a huge issue in the west, because it caused a rather large fight between the emperors and the popes, exacerbating existing divisions, eventually influencing the papacy's decision to side with the Carolingians, among other things.
A good read on the subject: