Really, I love George Washington. But I hate to let an opportunity pass to point out how self-abasement sometimes looks like humility (or humility sometimes results in unbiblical self-abasement).
Anyway, the prayer is "A Prayer For Wednesday Morning", and can be viewed in its entirety here (See #7).
The beauty and eloquence of the prayer are beyond dispute, but the theology leaves much to be desired. The baby of the New Covenant (regeneration and the New Creation) is thrown out with the bathwater of sins confessed.
When Washington says, "[I] humbly acknowledge the corruption of my nature...", he is using the language the Scripture uses of an unregenerate person, born in sin and unchanged.
He who is in Christ is a New Creation (2 Cor. 5:17), with a new nature (spirit) which is one spirit with Him (1 Cor. 6:17). The old man (nature) has been crucified with Christ.
This is why Paul, in Rom. 7, was careful to point out that the "sin" which dwells in our flesh, our very members, is not us ("I find that it is no longer I who do it, but sin which is in me.")
When Washington says, "[I] desire to be vile in my own eyes, as I have rendered myself vile in thine," he will be hard-pressed to find any such thought in the New Covenant epistles, since the child of God has been declared righteous, and has the awesome love of God promised in such a way as that *nothing* shall separate us from it. Not exactly "vile" in the eyes of Him who has given us His very righteousness.
Of course, we certainly were vile in His eyes before He gave us a new heart, as promised in His prophecies of the New Covenant. "...such WERE some of you. But you were washed, but you were sanctified, but you were justified in the name of the Lord Jesus and by the Spirit of our God." (1 Cor. 6:11)
Washington's thinking on this, which is all too common, is actually destructive to walking by the Spirit, which is why Paul gives the foundational imperative in Rom. 6:11, "Reckon [consider, choose to believe] yourselves to be dead to sin, but alive to God in Christ Jesus."
If someone thinks of themselves as still "the Old Man", even if saved by grace, they will be more inclined to act like the Old Man.
If you are a believer in Jesus Christ, you have gone from being "vile" in His eyes, to being "the apple" of His eye. Bask in that, and follow Him.