This story occurs in the universe of s/5782108/1/Harry_Potter_and_the_Methods_of_Rationality, diverging in Ch. 89. It contains spoilers for that story through that chapter, and is unlikely to make much sense without having read the same.
I would expect this to be the first (deeply ingrained) response of anyone who grew up in a first-world country. It's so annoying to be in the position of yelling at a character to do something obvious...
"Bring her back."
"Harry -" the old wizard began. His voice cracked. "Harry -"
"Have Fawkes cry on her or whatever. Hurry up." The voice that spoke sounded perfectly calm.
"I, I can't, Harry, it's too late, she's dead -"
"So there's nothing you or Madam Pomfrey or St. Mungos can do?"
"There isn't anything anyone can do! Her soul has departed, she's passed on!"
"Fawkes, help me!" Harry held Hermione's shoulders and pictured in his mind, as well as he could remember, the location of Radcliffe Hospital, the only hospital with which he was familiar.
An instant later, he pushed through the front doors of the ER, half carrying and half dragging her, and yelled as loudly as possible "Help me! She's not breathing!"
Faced with such trauma, the medical professionals reverted to their training. A paramedic felt for a pulse and began CPR. Another rushed over with a stretcher, while a doctor called for plasma, O negative blood, and an open OR. As they rolled the stretcher to the operating room, the last thing Harry heard was the doctor calling for a shot of epinephrine.
"What happened?" the admitting nurse questioned Harry.
"Um... car accident... Um, is there a restroom? I think I'm going to vomit now." The nurse pointed. Harry turned and walked to it.
As he walked away, he heard another nurse ask the admitting nurse "Aren't you going to tell him he can't have a parrot in here?"
"Parrot? What are you talking about?"
"You didn't see that large red bird flying around him?"
Harry turned to Fawkes "stay with Hermione. Do what you can for her, and try to stay out of sight."
After doing his best to clean the blood off his hands and clothes, Harry transfigured his robes into jeans and a t-shirt so he would be less conspicuous. Then he put on the invisibility cloak and sat down outside the operating room; he was in no mood to answer questions or talk to anyone.
Later, back at Hogwarts, Dumbledore and McGonagall had brought the Grangers and Verres-Evans to his office. Which had taken an annoyingly long time with Fawkes absent. After the "Sorry for your loss" conversation, and summary of events, "So that's the situation. Do you have any idea where he would have gone?"
Michael Verres-Evans answered instantly "Obviously, a hospital. Since distance doesn't seem to matter to you people, he'd probably go to the one near Oxf-."
"No, we checked St. Mungos. No one there had seen Harry." He paused, thinking, then "Expecto Patronum. Harry, where are you?"
The Patronus returned. "Radcliffe Hospital, room 42" answered Harry's voice. Michael Verres-Evans stared at the wizard with undisguised contempt.
A short time later, the six of them arrived in a recovery room. Hermione had a breathing tube and an IV, and a monitor showed her heart beating steadily. Fawkes was singing to her. Harry got up from the chair and took a step back as her parents rushed over.
Dumbledore spoke hesitantly "She died Harry. I don't know what-"
Dr. Michael Verres-Evans interrupted authoritatively "Death isn't that simple. It used to be determined by whether someone stopped breathing or had no pulse, but those methods are unreliable. If death is defined as the state from which one can't recover, then what it means to be dead changes every year: an injury which would have killed someone in the past might be healed today. Others try to get around that issue by referring to people as having died and recovered, but that just raises the question of whether the word retains any meaning at all." Petunia had taken her husband's arm and was not-so-subtly trying to pull him away from Dumbledore.
Dumbledore continued as if he hadn't been interrupted "I don't know what those Muggle machines are doing or what you think you've accomplished. Hermione is gone. She's not going to wake up, she's not going to return to Hogwarts, and everything is not going to go back to the way it was before."
"So that's a testable prediction: you don't expect her to wake up. Therefore if she wakes up then it implies that one of your beliefs is necessarily wrong, either your belief that souls exist or your belief that they can't reattach to a body after you've decided that they left or something like that. That things aren't going to be the way they were is obvious. Whether she goes back to Hogwarts depends on her and your decisions after she wakes up, I guess."
There was silence for a moment while Harry and Dumbledore stared at each other. Then Petunia tried to change the subject "I only saw a little bit of your school, but it didn't look very wheelchair accessible."
"I have a flying machine in my pouch right now, Mom. I really hope that's the most serious problem she has in the near future."