That leaves it unexplained then why David said his sin was against God alone.David was guilty of adultery and murder. Scripture is clear. (2 Sam. 12:5-9)
Stranger
The effect was the same but David did not kill Uriah using a sword. And it was Uriah's ex-wife he had sex with. The words murder and adultery are not used.
Nathan's criticism is more about David's not behaving a king anointed by God should have.
1:13 And David said unto Nathan, I have sinned against the Lord. And Nathan said unto David, The Lord also hath put away thy sin; thou shalt not die.
He should have had Uriah put on trial for blasphemy. Instead David wanted to hush the matter up, so he had Uriah put in the front line at a battle. Why blasphemy? For swearing by the life of the king, and since God had picked David as king, Uriah was guilty of blasphemy and should have been tried in a court of law. David perverted justice.
11:11 And Uriah said unto David, The ark, and Israel, and Judah, abide in tents; and my lord Joab, and the servants of my lord, are encamped in the open fields; shall I then go into mine house, to eat and to drink, and to lie with my wife? as thou livest, and as thy soul liveth, I will not do this thing.