Yes, he did.
The Romans were experts at this and knew exactly what it took to kill someone and what it took to make that process slow. Crucifixion was designed to drag the act of dying out over a long period of time.
As you note in the John passage, Jesus died more quickly than they expected. The trial and execution had been rushed into an abnormal timeline already, the Sabbath was coming1, there had been an earthquake2 and they had other things they needed to attend to. In order to be sure that the death sentence was carried out without any chance of a criminal being revived after a "mostly dead all day"* experience the soldiers were ordered to break their legs1. This would accelerate the process from long hours and even days to a matter of a few minutes as the victims would suffocate: unable to pull push themselves up to catch a breath of air.
When they came to Jesus they didn't bother because he was already dead.3 Just to be sure they checked with a spear in his side (probably just under his rib-cage piercing a lung) and blood and water came out4. The fact that liquids has begun to separate is a sure medical indication of death.
This whole process fulfilled several prophecies. First of all he was pierced, something that was prophesied about his death5. Secondly not a bone was broken. Crucifixion didn't usually break any bones unless they did the little acceleration trick, which was ordered for him, but it was also prophesied that he would not suffer broken bones6. See also: Did the Romans not breaking Jesus' legs fulfill a prophecy?
Lastly his unexpected early death shows that it was likely not actually the process of crucifixion that killed him, but that even at the very end he willingly surrendered his own life after having born the wrath of his father and completing the course of punishment that he had to endure. Once that was over, he willingly spent his last breath and surrendered his life just as he said he was going to do.
John 10:17 (ESV)
For this reason the Father loves me, because I lay down my life that I may take it up again.
* I need to ask on our meta The Princess Bride is an acceptable reference work.
John 19:31 (ESV)
Since it was the day of Preparation, and so that the bodies would not remain on the cross on the Sabbath (for that Sabbath was a high day), the Jews asked Pilate that their legs might be broken and that they might be taken away
Matthew 27:54 (ESV)
When the centurion and those who were with him, keeping watch over Jesus, saw the earthquake and what took place, they were filled with awe and said, “Truly this was the Son1 of God!”
John 19:33 (ESV)
But when they came to Jesus and saw that he was already dead, they did not break his legs.
John 19:34 (ESV)
But one of the soldiers pierced his side with a spear, and at once there came out blood and water.
Isaiah 53:5 (ESV)
But he was pierced for our transgressions;
he was crushed for our iniquities;
upon him was the chastisement that brought us peace,
and with his wounds we are healed.
John 19:36 (ESV) (This verse in turn quotes Psalms 34:20)
For these things took place that the Scripture might be fulfilled: “Not one of his bones will be broken.”