Why did he say "Lecha Levad Chatasi"?
Rashi: Therefore You can pardon. What I did evil to Uriyah, was evil only to You, for You commanded about it 1 .
Radak: The sin with Bas Sheva was covert; no one knew except for You. Even though he called for her via his servants, they did not know why he called her. Also, Uriyah, no one knows why I caused his death. Even though David wrote to Yo'av [to cause his death], people thought that it was because Uriyah transgressed the king's word.
Radak citing Rav Sadya Gaon: I confess only to You 2 .
Radak citing his father: If Uriyah were alive, the sin would be against You and him. Now that he is dead, I sinned only against You. There is no one else for me to request pardon from him.
Sha'arei Teshuvah (1:45) #1: I need only Your pardon. If I sinned against a person, I already requested pardon from him.
Sha'arei Teshuvah (1:45) #2: I did not sin against a person; I did not request excessively, nor steal anything that I must return. My Kaparah depends only on Your pardon.
Malbim: I did not sin against Uriyah -not for killing him, for he was Chayav Misah for Mored b'Malchus, and not for Bas Sheva, for she was divorced 3 , like I explained in Shmuel II, Perek 11.
Why is this unlike other sins Bein Adam l'Chavero, which require the victim's pardon? Refer to 51:6:1:4, 7 and the note there. (PF)
Radak: A support for this is Yuma 86b, which calls Arayos Bein Adam la'Makom, so it is proper to cover up such sins (refer to 32:1:5:2). One who publicizes his sins is brazen (Brachos 34b).
Even if she was divorced, Nasan's parable of stealing the Oni's lamb implies that David sinned against him (Uriyah was destined to remarry her after the war)! Perhaps since Uriyah did not return after the war, retroactively David did not sin against him. He sinned against Hashem for killing him via a ruse. (PF)
Why did he say "veha'Ra b'Einecha Asisi"?
Malbim: The sin was only against You; if it were in public, there would be Chilul Hashem, and Teshuvah would not help.
What is the meaning of "Lema'an Titzdak bi'Dvarecha"?
Rashi (from Sanhedrin 107a): I had strength to overpower my Yetzer; I did not, lest people say 'the Talmid defeated his Master 1 .' I told You "Bechaneni Hashem v'Naseni" (26:2). You tested me, and I was not complete, so You will be correct, and not me.
Rashi: If You will pardon me, You will be justified in Your judgment against all Resha'im who do not repent. They will not be able to say 'had we repented, it would not have helped.'
Radak: You will bear my sin, like You told Kayin "ha'Lo Im Teitiv Se'es" (Bereishis 4:7), and when Moshe asked to know Your ways, You said "Nosei Avon va'Fesha v'Chata'ah" (Shemos 34:7).
Sha'arei Teshuvah (1:45): In order to show nations Your great Tzedek and forgiveness on the day that You judge me. It says "Lema'an", for via the magnitude of the sin, this shows Your great Tzedek in pardoning.
Ramah (Sanhedrin 107a): At David's level, he was not proper to sin; he should have defeated his Yetzer. However, he thought that he was at the level of the Avos, and asked to be tested. This was against Hashem, who knew that he was not at the Avos' level, so He did not want to test him. Since David entered this against Hashem's will, this aroused much opposition 2 , and caused that he not defeat his Yetzer, lest people say 'the Talmid defeated his Master.'
Malbim: "Lema'an" applies to "Kabeseni me'Avoni" (verse 4). If You will not pardon me, people will say that You are not conducting with Tzedek and Mishpat.
This implies that David intentionally sinned, lest people say
What is the meaning of "Tizkeh v'Shaftecha"?
Radak: [Via pardoning me,] You will merit in Your judgment. One will not be able to say that You said one thing, and did differently; You said that You bear sin of sinners who repent.