"The Paralympic Games is about transforming our perception of the world. We are all different, there is no such thing as a standard or run-of-the-mill human being, but we share the same human spirit," said Hawking, who was given two years to live in 1963 after he was diagnosed with motor neuron disease.
"What is important is that we have the ability to create ... however difficult life may seem, there is always something you can do and succeed at," he added.
The extravaganza, directed by Bradley Hemmings and Jenny Sealey, was billed as a voyage across "a sea of ideas" including Isaac Newton's discovery of gravity, the force that all Earth-bound athletes strive against. The show included 73 deaf and disabled professional performers and 68 disabled people among its 3,250 volunteers.
Sebastian Coe, chief of the London organizing committee, issued a big welcome home "to a movement that shows what sport is all about."
"Sport is about what you can do, what you can achieve, the limits you can reach, the barriers you can break. Sport shows what is possible. Sport refuses to take no for an answer," Coe told the audience of 60,000.
The London event is on track to be the most-watched Paralympics ever, with 2.5 million tickets expected to be sold by the time it ends Sept. 9.
As the athletes paraded in under a full moon, a huge roar filled the stadium for South African flagbearer Oscar Pistorius, the sprinter who is making history by running in both the Olympics and the Paralympics this year. Glittery ticker tape and a standing ovation then greeted the enormous British team as they entered to the David Bowie song "Heroes."
The parade took two hours nearly an hour longer than expected but the joy and pride that accompanied it was stirring to watch. Some athletes came in motorized carts, others propelled themselves in their wheelchairs, still others were pushed by coaches or volunteers. Athletes walked in with canes or crutches, eye patches and sunglasses, prosthetic limbs, no limbs and walking sticks, determined to make it around the imposing stadium to a global music mash-up by local DJs.
In a nod to the famously erratic British weather, umbrellas were a central theme. Seeing performers with no legs beneath the knee doing aerial flips while carrying umbrellas could inspire the most ardent couch potato.
After blind soprano Denise Leigh sang the tribute song "Spirit in Motion," several Paralympians took flight in an elegant, slow-motion aerial ballet.
And of course, this being Britain, the words of Shakespeare once again made an appearance, with both Miranda of "The Tempest" and British actor Ian McKellen announcing that "the greatest adventure is what lies ahead."
That, over the next 11 days, includes Paralympic athletes competing in 20 sports, including archery, cycling, rowing, equestrian, sailing, sitting volleyball, wheelchair rugby, wheelchair tennis and wheelchair basketball.
At the start, Hawking directed the fictional Miranda to "be curious" and the stadium was transformed into a giant blinking eye, with performers on huge waving sticks acting like eyelashes. Along her travels, Miranda navigated a maze to find an apple and everyone in the stadium took a bite out of their apple along with her.
Outside the stadium, hundreds of performers cheered, sang and danced. Jason Kajdi, an 18-year-old from south London, did huge splits with bouncy legs that resembled the "Cheetah" prosthetic limbs worn by Pistorius.
"Never used these before this," Kajdi admitted. "They are brilliant fun but hard work."
Authorities, meanwhile, promised to provide "a grand and global stage" for a games that everyone will remember.
Queen Elizabeth II formally opened the games, saying the nation looked forward to "celebrating the uplifting spirit which distinguishes the Paralympic Games from other events." She was accompanied by her grandson, Prince William, his wife Kate, the Duchess of Cambridge, and Prime Minister David Cameron.
The Paralympic movement tracks its beginnings to the vision of Dr. Ludwig Guttmann, who in 1948 organized an archery competition for 16 injured patients at Stoke Mandeville Hospital in Britain.
Wednesday's ceremony concluded with a bevy of fireworks and Miranda breaking a glass ceiling just as Paralympians must smash through their own barriers. Margaret Maughan, who won Britain's first gold medal at the 1960 Rome Paralympics in archery, then lit the cauldron.
Society, too, was encouraged to abandon old-fashioned perceptions of what disabled people can and cannot do.
Pistorius, at an earlier news conference, praised London for its inclusivity.
"Kids didn't stare at people's prosthetic legs and they were asking guys in wheelchairs what events they do," Pistorius said. "There are a lot of people here who don't focus on the disability anymore they focus on the athletes' abilities ... there's nothing to be ashamed of."