In scripture, the Lord expresses a certain appreciation for those who have clean hands.
The righteous also shall hold on his way, and he that hath clean hands shall be stronger and stronger. (Job 17:9 KJV)
He that hath clean hands, and a pure heart; who hath not lifted up his soul unto vanity, nor sworn deceitfully. He shall receive the blessing from the LORD, and righteousness from the God of his salvation. (Psa 24:4-5 KJV)
Scripture also reveals the Lord's condemnation of those who have sullied their hands by harming others.
These six things doth the LORD hate: yea, seven are an abomination unto him: A proud look, a lying tongue, and hands that shed innocent blood, (Pro 6:16-17 KJV)
A closer look at the scripture reveals that the Lord offers to give us clean hands through confession, not by any cleansing we might do ourselves. Indeed, scripture cites trying to cleanse ourselves of iniquity as futile.
Even if I washed myself with the strongest soap, God would throw me into a pit of stinking slime, leaving me disgusting to my clothes. (Job 9:30-31 CEV)
In both Old and New Testament, the Lord insists we can only gain the cleansing we need by asking for his provision whenever we need forgiveness. These scriptures make no allusions to reliance on our own efforts. He knows we can't cleanse ourselves.
Remove my sin, and I will be clean; wash me, and I will be whiter than snow. (Psa 51:7 GNB)
Come close to God and He will come close to you. [Recognize that you are] sinners, get your soiled hands clean; [realize that you have been disloyal] wavering individuals with divided interests, and purify your hearts [of your spiritual adultery]. (Jam 4:8 AMP)
While we shouldn't give up hand-washing, it seems prudent that we think about our reliance on God's forgiveness when we wash. As a means of avoiding the sin of self-righteousness, this may become a good defense against thinking we can cleanse ourselves. Remembering God's grace when we wash may also deepen our relationship with the God of forgiveness. Unlike the Pharisees and Saducees of Jesus' day. we know that relying on God's forgiveness frees us from ritual cleansing and puts our relationship with Christ in the center of our thoughts.
*Zhong, C-B. & Liljenquist, K. (2006). Washing away your sins: Threatened morality and physical cleansing. Science, 313, 1451-1452.