I believe that murder is a crime against God and against man simutaneously, and reconciliation must be sought with both. The "my sin is private" thinking that has gained a foothold in much of Christianity flies right in the face of Scripture.
ALL sin is against God and man, whether it be another or oneself.
When Paul addressed sexual immorality within the community of Christians, he treated it as a crime against the Church, defiling the whole body.
Methinks your interpretation of the passage referred to is flawed. Paul's emphasis is upon one's own physical body and the soul, which are created in the image of God and thus this particular sin, sexual immorality, is a heinous sin indeed.
So let me turn this back on you. Are you suggesting that if someone commits murder, they may conceal their crime from the justice system so long as they confess it to God privately?
1. You haven't address my question(s) to you which focus upon the sufficiency and legality of Christ's atonement. Will you please do so?
2. To answer your question in a general way, my answer would be, "No." Where possible, reconciliation with an offended party should be pursued. However, in the case of murder, it is hardly possible to make amends with a victim who is dead. In your other post you offered a situation where a murderer hides his sin for a period of time but then later confesses that sin to the authorities and then remarked that due to the fact that the crime was hidden for a time, even though later confessed, this person could never receive mercy from God. Thus my question(s) to you for the purpose of soliciting clarification and your view. Please do so.