Christmas and the cross must go together. Without the cross, the manger is essentially meaningless.
In his book, Whatever Happened to the Gospel of Grace?, James Montgomery Boice writes,
Christmas by itself is no gospel. The life of Christ is no gospel. Even the resurrection by itself is no gospel. The Good News is not just that God became man, nor that God has spoken in Christ to reveal a proper way of life for us, nor even that death, our great enemy, has been conquered. The Good News is that sin has been dealt with, that Jesus suffered its penalty for us as our representative, and that all who believe in him can look forward confidently to heaven” (p.105).
The story of the incarnation of Christ that we rightly focus on every year at Christmas, as wonderful as it is, saves no one apart from the cross. The life of Christ, as important as it is, saves no one apart from the cross. Christ certainly calls His people to follow Him (Mark 1:17; 8:34), and so to obey His commands and to emulate His example, but without the cross, the “right way of living” saves no one – it would still just lead to death.
The purpose of Christ’s incarnation was so that He would save His people from their sins (Matthew 1:21), and that means, first and foremost, that He was born so that He might die in the place of sinners.