Speaking at the offices of think-tank Policy Exchange, Mr Gove said he was running for leader due to "conviction" rather than "calculation or ambition".
Justice Secretary Michael Gove will make a speech later after his last-minute decision to enter the race stunned Westminster and forced his Vote Leave partner Mr Johnson to drop out.
Britain's Daily Mail newspaper on Friday backed Theresa May's bid to become the next prime minister, saying she was the only one out of five candidates for the Conservative Party leadership who was up to the job.
"I did not want it, indeed I did nearly everything not to be a candidate for the leadership of this party", Gove said in a speech in central London as he launched his campaign. "The country voted for no more politics as usual, no more business as usual".
"I did not want it, indeed I did nearly everything not be a candidate for the leadership of this party".
Living up to his reputation as a political maverick, the former London mayor outlined the demands of the role over the course of a lengthy speech in London and then said: "Having consulted colleagues and in view of the circumstances in Parliament, I have concluded that that person can not be me". "My judgment about what is right for this country will always guide me and on that basis I came reluctantly and with a heavy heart".
Gove said he was best-placed to succeed Cameron because he campaigned to leave the European Union, which Britain is now committed to doing after 52 percent of voters backed quitting the bloc in last Thursday's momentous referendum.
The justice secretary stated: "This country voted for change - and I am going to deliver it".
Boris Johnson's allies accuse Michael Gove of 'midnight treachery'
"Having consulted colleagues and in view of the circumstances in Parliament, I have concluded that person can not be me", he said to the astonishment of journalists and supporters.
The race to become Britain's next prime minister took a dramatic, unexpected turn Thursday as former London Mayor Boris Johnson - popular with the public and widely considered to be a front-runner - ruled himself out of contention after the defection of a key ally.
The victor will be announced September 9, and will face the challenge of starting British exit negotiations with the EU.
The Liberal Democrat leader, Tim Farron, said Johnson had followed "the Bullingdon playbook: you break it and someone else has to fix it", referring to the Oxford drinking society of which Johnson and David Cameron were members.
He said: "Theresa is the best person to lead our exit from the European Union so that we reduce immigration and regain sovereignty while protecting our hard won economic growth".
"We are paying for storage facilities for these water cannon - that beggars belief", the Guardian reported Khan to have said.
By leading the United Kingdom out of the EU, Gove promised to end free movement and to introduce an Australian-style points-based system for immigration, and bring numbers down.