Speaking at London's Science Museum the scientist also says he believes the future of humanity depends on living on other planets.
"The long-term future of the human race must be space and that it represents an important life insurance for our future survival," he says.
"It could prevent the disappearance of humanity by the colonisation of other planets."
Professor Hawking made the comments while escorting an American visitor around the museum as part of a guest of honour prize.
"Sending humans to the moon changed the future of the human race in ways that we don't yet understand," he went on.
"It hasn't solved any of our immediate problems on planet Earth, but it has given us new perspectives on them and caused us to look both outward and inward."
Adaeze Uyanwah, 24 and from California, won the tour after producing a blog and video describing a "perfect day" in London.
She asked Prof Hawking what human failings he would alter.
"The human failing I would most like to correct is aggression," he says.
"It may have had survival advantage in caveman days, to get more food, territory, or partner with whom to reproduce, but now it threatens to destroy us all.
"A major nuclear war would be the end of civilization, and maybe the end of the human race."
The scientist then got a bit spiritual when he told her: "The quality I would most like to magnify is empathy. It brings us together in a peaceful, loving state."
Ms Uyanwah says she was blown away by the meeting.
"It's incredible to think when my grandchildren are learning Stephen Hawking's theories in science class, I'll be able to tell them I had a personal meeting with him.
"It's something I'll never forget."
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Stephen Hawking says space travel will save the human race.