Continuity mistake: Right after John Smith gets captured and is taken away, Nakoma grabs Pocahontas' hand, but when it shows the shadow of them they aren't holding hands. Then in the very next shot they are but it doesn't show them grabbing each other's hand again.
Continuity mistake: When Pocahontas sees John Smith for the first time, John is climbing up a cliff face and then a tree. You can see his musket strapped to his back as he's climbing. But after he tells Thomas he's getting a better look and then is shown in the tree, his gun has vanished.
Answer: This is a question the movie chose not to answer. It could be assumed he died as the Powhatans did not have the knowledge to treat gunshot wounds. However, there were cases even as far back as this time of individuals being able to survive gunshot wounds as long as the bleeding could be stopped and infection didn't spread. As the Powhatans did know herbs in the land to treat infections and did know how to mend bleeding; it is also possible he lived though he would have walked with a limp for the rest of his life.