The family's attorney says the man was taken off life support and died on Friday morning.
The Spokesman-Review reported that the man was identified by friends of the family as Will Berger. The paper reported he was in his mid-30s, and apparently had suffered a head injury just days before was still wearing a hospital bracelet when he began to act irrationally.
The case began Thursday at Oz Fitness on Spokane's South Hill, where the man was lifting weights in the gym.
Witnesses say he started throwing weights and screaming, and he punched a hole in a wall and knocked down a paper towel dispenser.
He ran out of the gym into traffic and was confronted by deputies who shot him twice with a stun gun. The man's heart stopped, but he was revived and taken to a hospital.
The State Patrol is investigating.