Of course King David was not alive to see the crucifixion of Jesus.
"About the ninth hour Jesus cried out in a loud voice, “Eli, Eli, lema sabachthani?” which means, “My God, My God, why have You forsaken Me?” (Matthew 27:46)
Some saw the righteous one forsaken, but three days later the one forsaken righteous human was raised to life, never again to be forsaken - and the judgment of all other humans has been given to Him.
"And I have a hope in God, which these men themselves also accept, that there is going to be a resurrection, both of the righteous and the unrighteous." (Acts 24:15)
"Father forgive them, they do not know what they are doing." (Luke 23:34)