The MP for Penrith and The Border quit the Conservative Party last week and said he planned on standing in the capital as an independent candidate.
Mr Stewart, who stood in the recent Tory leadership contest, said as a constituency MP he did not have real power to change the lives of the people he represented.
"The reason I want to get back to local politics is that one of the problems as a constituency MP is that what your constituents are asking you to do are often what you have no power over," he said.
"You often end up writing letters asking favours from everyone. What they actually want you to do is sort out the police, sort out the transport and housing - these are powers of local government.
"The healthiest bit of British government is local government and that's why I want to get out of Parliament. I'm leaving Parliament. I don't feel this is an easily fixed system."
He was speaking at an event at the Cheltenham Literature Festival with fellow MP Jess Phillips, The Times columnist Philip Collins and Lord Cooper, the founder of polling company Populous, discussing whether it was the end of political parties in the UK.
Mr Stewart said that despite the Conservative and Labour parties both moving away from the centre of British politics, he still believed there was a middle ground.
"My feeling is that in my lived experience we are not a country that divides into two hostile camps," the former international development secretary said.
"My faith in this system is being tested to the limit but I still believe that neither of these parties is going to be able to win a majority until they move back into the centre ground."
He was asked whether he thought he could do what Emmanuel Macron did in France when he ran for President by founding his own centrist movement En Marche.
"The big choice for someone like me is whether to just say I am an independent for Mayor of London or whether to try and construct a new party," he said.
"I think, particularly for a mayoral race, being an independent is a very strong position to be in.
"The problem for the Labour and Conservative candidates is that you spend your time apologising for Jeremy Corbyn and Boris Johnson.
"I think there is something really powerful about being yourself and trying to be honest about what you believe."
He said he was "interested" in exploring whether voting should be made compulsory, like in Australia.
"I'd like to see a push towards requiring 100% of people to vote so that our Members of Parliament can be genuinely representative of everyone, particularly the poorest," he said.
And he said that with the current political class, "rhetoric has come completely apart from reality".
"There is only one answer for this, only one way of overcoming these extreme divisions, only one way of overcoming the logic of snake oil salesmen and fairy story purveyors, which is in the end the courage connected with leadership," he said.
"Only through leadership and taking the risk of being honest and not caring or not whether it's popular, not really caring whether you can sell fairy stories and nonsense to get votes and heading to the centre ground, are we going to save the country."