unthinkable and unspeakable.  If Linda had found that she was pregnant while in the World State, she could simply have had an abortion and all would have been well.  But she was not in the World State and ended up having John.  Because she had him, she felt she could not go back to civilization in her youth.  As she tells John inChapter 8,

Having young ones like an animal … If it hadn't been for you, I might have gone to the Inspector, I might have got away. But not with a baby. That would have been too shameful.

Having John trapped Linda on the reservation.  While she was trapped there, her life changed.  She essentially lived by selling her body to men.  Then she got fat and lost teeth and generally became physically unattractive.  When she finally returned to the World State, the shock of being back in civilization was too much for her since she was a freak to the people there.  She was old-looking, fat, and worst of all a mother.

Thus, having John essentially doomed Linda and he can be said to have caused her death.

John, the Savage, hangs himself at the end of the novel. He had previously asked Mustapha Mond if he could be sent away to the islands with Helmholtz. Mond, however, refused:

“He said he wanted to go on with the experiment. But I’m damned,” the Savage added, with sudden fury, “I’m damned if I’ll go on being experimented with. Not for all the Controllers in the world. l shall go away to-morrow too.”

“But where?” the others asked in unison. The Savage shrugged his shoulders. “Anywhere. I don’t care. So long as I can be alone.”

John doggedly pursues his dream of a life alone, finding a lighthouse and buying seed so he can grow crops. But crowds find him. He whips himself because he believes this will purify him and atone for his sin, and in a society built on the absence of pain, this makes him a media spectacle. He becomes the stuff of newspaper headlines -- and even the subject of a film -- because "pain was a fascinating horror." Newspaper helicopters hover, and hordes of people come to his lighthouse, chanting for him to whip himself.

The implication is that John kills himself because he can find no other way to be alone, no other escape from a society in which he has nothing in common. Who bears responsibility for his death? John bears some responsibility, as he makes the choice. The crowds that won't leave him in peace bear some of the blame, too, though they can hardly be expected to understand what they are doing to John. Perhaps more responsibility lays with Mond, who has both the intelligence and the power to help John but chooses to treat him as an experiment, not a human being.

I suppose that depends on how you define "responsible for."

As far as how John the Savage dies, he kills himself.  Specifically, he hangs himself.  So in that sense, he is responsible for his own death as no one helps him die or causes him to die.

If you want to assign moral blame, I suppose you could say the crowd of people (and the society as a whole) is responsible.  They are the ones who start chanting "orgy porgy" as John whips himself.  They are the one who start the orgy that makes John so disgusted with himself.  So you could say that it is they (because of the difference between their values and his) that cause John's death.