London mayor Boris Johnson said: "At last, justice and the well-being of Mr McKinnon have prevailed.
"I have long supported Gary's right to be tried here in the UK.
"To extradite a man diagnosed with Asperger syndrome to America for trial would have been extraordinarily cruel and inhumane. I applaud the Government's stance.
"If they had approved extradition, they'd have been saying that extradition on any grounds was OK. It's not."
Melanie Riley, of campaign group Friends Extradited, said: "We are relieved and delighted for Gary and his family and his lawyers.
"A brave and correct decision has finally been made by the Home Secretary."
More top news
For thousands of families, today is the day they find out whether their 11-year-olds will go to the secondary school of their choice.
A dangerous "charmer" who murdered his girlfriend over an obsession she was having an affair on Facebook has been sentenced to life.
EastEnders star Danny Dyer is set to marry his long-term girlfriend after she proposed to him ... and Katie Hopkins isn't happy.