Owen Smith said “something had gone badly wrong” since Mr Corbyn took charge amid fresh claims of abuse being directed at critics of the leader by activists.
Mr Corbyn insisted he is not a bully and said he was “disappointed” at the claims made against him, which included a suggestion he threatened to ring the father of a critical MP in order to put pressure on him.
Former shadow cabinet minister Mr Smith warned the party could be “destroyed” and “consigned to history” unless it could unite.
Mr Smith told Sky News he had received death threats and the problem of abuse had not been there before Mr Corbyn’s leadership win.
He accepted he had “never been bullied by Jeremy” personally but “under his leadership, there has been a culture of bullying, I fear”.
He added: “Jeremy, of course, always says that he doesn’t condone it but somehow under his leadership - we can’t deny the facts that this wasn’t something that we saw in the Labour Party before Jeremy Corbyn became leader - and it’s now become commonplace in the Labour Party. So something has gone badly wrong under his watch.”
Mr Smith also hit out at the suggestion Labour MPs will face having to be re-selected to stand in the 2020 general election - a move which could allow Mr Corbyn’s supporters in the grassroots to oust critics.
He said: “It’s not kinder and gentler, is it, if you are the boss of an organisation and the workers are unhappy, to threaten to give them the sack. It’s the sort of thing you might see at Sports Direct but it’s not what you should be expecting in the Labour Party.”
The party leader dismissed concerns about the prospect of re-selection hearings, insisting “it is not Armageddon” and was simply the result of planned boundary changes as part of the Government’s efforts to cut the number of Commons seats from 650 to 600.
“There will be a process by which MPs will be automatically short-listed for the new constituencies because they are sitting MPs and because they have a geographical interest in the new constituency,” he told BBC Radio 2’s Jeremy Vine.
It was up to local constituencies to decide whether they then wanted an open selection process.
Mr Corbyn is favourite to win the postal ballot of Labour’s members - whose ranks he said have swelled to more than 500,000 - as well as the 183,000 people who signed up this week as registered supporters and the affiliated supporters in the unions.
Addressing allegations of abuse from former leadership contender Angela Eagle and party whip Conor McGinn, he told Sky News: “I wish some of my colleagues would concentrate on political issues. I regret the language that’s been used, by all of them.
“I don’t do any abuse, I don’t do any bullying, I don’t allow it to be done anywhere to do with any of my campaign teams and I’m very surprised and very disappointed they should say that because politics has to be about bringing people in.
“I think we have done that spectacularly well - we now have the largest membership we’ve ever had. That’s good, that means more and more people are involved in politics.
“That’s good, it’s not a threat, it’s a good thing that people come together and want to debate and be active in politics in our society. Isn’t that good for democracy?”
His comments follow an extraordinary claim from Mr McGinn, who said Mr Corbyn considered calling his father - a Sinn Fein councillor - in an effort to “bully” him following critical comments the MP made in a magazine interview.
Mr Corbyn’s office dismissed the allegation as “untrue” but St Helens North MP Mr McGinn accused the party leader of hypocrisy for talking about a “kinder, gentler politics” when “he had proposed using my family against me”.
He said: “The leader of the Labour Party was proposing to address an issue with one of his own MPs by ringing his dad.”
Mr McGinn said: “I am afraid I could no longer tolerate the hypocrisy of him talking about a kinder, gentler politics when I knew for a fact that he had proposed using my family against me in an attempt to bully me into submission because he didn’t like something I said.”
In a statement, Mr McGinn said he and other Labour MPs had been subjected to a “torrent of abuse and threats” from supporters of Mr Corbyn.
“In my constituency, a group of people gained access to my shared office building under false pretences and filmed themselves protesting outside the door of my office, in an incident that has been reported to the police,” he said.
“They threatened to disrupt my surgeries and events I was attending, requiring me to have a police presence at those last weekend.”
Former shadow cabinet minister Ms Eagle, who launched a leadership challenge against Mr Corbyn before standing aside to give Mr Smith a clear run, suggested the leader had contributed to abuse directed at her and her staff.
She has cancelled constituency surgery meetings, a brick was thrown through her office window and her local Wallasey constituency Labour Party has been suspended amid bullying claims.
Ms Eagle told the Daily Telegraph: “I think he has contributed to this. It’s all very well to condemn it but there’s a permissive environment. You can make any number of ritual condemnations as you like but you have got to be judged by your actions, not just words.
“He has been stirring, he needs to be held to account. We have contacted the police and they have said we should cancel surgeries for safety reasons.”