Q: Did God die on the cross?
A: First, it seems important to define death. According to Princton WorldNet, dead means “no longer exerting force or having energy or heat.”
Knowing that Jesus is fully God and fully man, two distinct and inseparable natures in one person.
Jesus died. Since His natures are inseparable, it is clear that both his human and divine natures experienced death.
The problem with the question is that it connotes the entirety of God. First, Jesus is one person of the Godhead. While Jesus was in the grave, the Father and the Holy Spirit were not under the power of death. Second, Jesus took for himself a reasonable soul as well as a body when he was incarnate (Matthew 26:38). Jesus’ soul, operating as a human soul (as it is a human soul), does not cease to exist. Rather, Jesus’ soul continued past death just as a human soul is expected in the Christian Worldview. His soul descended into the place of the dead and was with His Father the very day of His death.
Did God die? Yes. But, be sure you correctly understand death and how it affects the body and soul.