Washington has no conversion story. In John 3:3 Jesus said "Truly, truly I say to you, unless one is born again he cannot see the kingdom of God." Washington evidences no time period in his life where there is a change comparable to what Jesus talks about here. We have many of his diaries and letters and personal letters and while he often speaks of "Providence", he never references a time when that vital change from life to death occurred in his own soul by believing on the blood of Jesus.
Washington never confessed Jesus before men. In Matthew 10:32 Jesus promised "everyone who acknowledges me before men, I also will acknowledge before my Father who is in heaven, but whoever denies me before men, I also will deny before my Father who is in heaven." The reason why a book had to be written defending Washington as a Christian is because he never acknowledged Christ before men. Washington would often acknowledge Providence (pp125, 148) and "that Almighty Being who rules over the Universe" (pp216) and "The Greatest and Best of Beings" (pp172). While each of these are valid titles for God, it is sad that Washington's thoughts were lacking of the Person of Christ. The Savior of the World, our Mediator, our Brother and our King is left out of Washington's discussion of God and this is the most worrisome fact about Washington's spiritual life.
Washington was a moral man. It is true that Washington conducted himself in a notoriously upright manner. Abigail Adams (John Adams wife) said of him "He is polite with dignity, affable without familiarity, distant without haughtiness, grave without austerity, modest, wise, and good" (pp228). One of his aids said "a complete knowledge of his honesty, uprightness, and candor in all his private transactions have sometime led one to think him more than a man" (pp.186). Comments like these and many more have rightfully left a great imprint in how we think of Washington. They inspire us to be great. Some scholars have even argued that the executive branch was crafted with his extreme moral responsibility in mind. Washington even had written a set of guidelines to help him in his daily interactions which you can find here. Yet, these resolutions are starkly contrasted with Jonathan Edward's which were full of God and faith and a devotion to his creator and savior.
Conclusion: Washington was a great moral example, and there is evidence that he went to church and mentioned a Higher Power on different occasions. But did he love Jesus? Did he see Jesus as the source of all his success and morality and life? The answer to these questions is sadly no because he never gave Jesus as much credit as "providence" or "that almighty being". If you know somebody you use their name and Washington rarely if ever used Jesus' name. When I try to summarize what was missing in Washington's spiritual life I think about what Paul talks about in 2 Corinthians 2:14-17:
"But thanks be to God, who in Christ always leads us in triumphal procession, and through us spreads the fragrance of the knowledge of him everywhere. For we are the aroma of Christ to God among those who are being saved and among those who are perishing, to one a fragrance from death to death, to the other a fragrance from life to life. Who is sufficient for these things? For we are not, like so many, peddlers of God's word, but as men of sincerity, as commissioned by God, in the sight of God we speak in Christ." (ESV)Amidst Washington's excellent ability to lead, exercise of great self control, and an inclination that there was a Higher Power who is leading this world forward he was sadly missing Jesus the Christ and therefore the true essence of Christianity.