Mr Umunna, 36, had been the second candidate to declare his intention to stand,via a post on his Facebook page.
But the MP said in a statement that he was not comfortable with the level of pressure and scrutiny that came with being a leadership candidate.
BBC correspondent Eleanor Garnier said it was a "huge shock" as Mr Umunna had been seen as one of the frontrunners.
She added that she understood there was no scandal that lay behind his decision to step aside.
Rather, Mr Umunna felt that due to the "sheer pressure" of the contest it was not the right time for him - or his family - to put himself forward.
'Further reflection'
A rising star of the party, Mr Umunna was first elected to Parliament in 2010 before joining the shadow cabinet.
He confirmed his candidacy for the Labour leadership on 12 May, saying the party under Ed Miliband had failed to appeal to enough voters in England.
But in a statement on Friday he announced his intention to withdraw from the race.
I understand that Chuka Umunna has told friends that "this is not right for me or people close to me. I'm human, the rest of my life is more important to me than politics."
The shadow business secretary has been distressed by the degree of scrutiny he, his family and his girlfriend have come under since he declared his intention to stand for the labour leadership.
I am told that this includes reporters waiting outside the houses of relatives late at night.
Friends of Mr Umunna deny that he is nervous of not getting the support of enough MPs to run for the leadership, they point out that he already has half the necessary names and that there are some weeks to go.
The shadow business secretary now wants to play a leading role in arguing that Britain should stay in the EU and does not rule out seeking the leadership in the future.
He said he thought he understood "the scrutiny and attention a leadership contest would bring", but added: "However, since the night of our defeat last week I have been subject to the added level of pressure that comes with being a leadership candidate.
"I have not found it to be a comfortable experience."
The shadow business secretary said he had wondered if it was too soon for him to stand for the leadership, and "I fear it was."
He also cited concerns about the impact on "the rest of one's life" and those close to him.
"Consequently, after further reflection, I am withdrawing my candidacy," he said.
Mr Umunna said he would continue to serve as shadow business secretary and hoped to "play a leading role" in Labour's campaign to keep the UK in the EU during the forthcoming in/out referendum.
Yvette Cooper, Andy Burnham, Liz Kendall and Mary Creagh have already thrown their hats into the ring for the Labour top job.
Candidates must secure nominations from 34 colleagues - 15% of the party's MPs - by 15 June to make it on to ballot papers, which will be sent to members in August ahead of the leader's election a month later.
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