Midrash su Salmi 51:3

And when Nathan the Prophet came and rebuked David for that act, he said (Ibid. 13) "I have sinned to the L-rd," and (Psalms 51:3) "Be gracious (chaneini) to me, O G-d, according to Your lovingkindness … (6) To You alone have I sinned and what is evil in Your eyes have I done."

"For You, You alone, have I sinned . . . That You may be justified when You speak" (Ps. 51:6). To whom may David be compared? To a man who broke a limb, and came to a doctor. The doctor marveled and said: "How great is thy break! I am much distressed on your account." The man with the broken limb said: "Art you distressed on my account? Was not my limb broken for your sake, since the fee is to be yours?" Just so David said to the Holy One, blessed be He: For You, You only, have I sinned: If You receive me, then if You say to transgressors "Why haven't you repented?" all transgressors will submit to You, for all of them will behold me [David], and I shall surely bear witness that You receive the penitent. Hence the Holy One, blessed be He, said: Behold, I have given him [David] for a witness to the peoples (Isa. 55:4). And God gave as a witness not only me, David, but all Israel, since it is said You are My witnesses, said the Lord, and My servant whom 1 have chosen (Isaiah 43:10).