Owen Smith said his "grave fear" is that the Labour Party will split unless Jeremy Corbyn is replaced as leader.
Speaking at the latest leadership hustings in Glasgow on Thursday night, Labour leadership challenger Mr Smith said: "I've never had any part in splitting the Labour Party.
"I've been Labour all my life and I'll be Labour till I die.
"My grave fear is that the party could split and we could end up with a divided party."
He said it was "essential" to guard against a split and added "to do that we need a new leader".
Asked about any potential party split, Mr Corbyn said: "It's an enormous honour to be selected and elected as a Labour Member of Parliament.
"You have a huge responsibility to the community that has elected you and the party that has selected you and put you there so the idea, as happened with the SDP in the 1980s, that you could walk away into the arms of wealthy donors and set up some other organisation is a complete and utter anathema to me.
"I want our party to be democratic and accountable."
On the issue of the Labour Party in Scotland, who were overtaken by the Tories at the Scottish elections in May, Owen Smith claimed the Labour Party in Scotland had gone backwards under Mr Corbyn's leadership.
He said: "People in Scotland can not look to Labour in Westminster right now and anticipate us winning a general election, therefore we are ill serving the Scottish Labour Party and we are ill serving the people of Scotland.
"We need to get back to being a radical and credible government-in-waiting, and we need to be much more powerful in attacking the Tories.
"The people of Scotland want a real opposition and the NHS is being privatised, houses aren't being built, our schools system is being wrecked across England, and Scotland is being underfunded."
In a sometimes ill-tempered debate Mr Smith went on to claim the Labour leader is "happy" about the Brexit vote.
He said: "The reason I think that Jeremy can be so complacent and passive about this is that he's happy about the result. He's not bothered about the result.
"If he's so concerned about protecting workers' rights why on earth would we allow the Tories to implement a Brexit deal which is going to see workers' rights in this country sold down the river?"
Mr Smith said he was "gutted" by the EU referendum result and was prepared to fight Brexit up to opposing a vote to trigger Article 50 in parliament.
He added: "He [Mr Corbyn] didn't fight hard last time round. I will."
Mr Corbyn said the referendum had taken place and delivered a result, which was not the one he wanted but it had to be respected.
He said he would fight to protect workers' rights and consumer rights and to ensure "there is an effective economic relationship with Europe when the Brexit negotiations are completed".
He added: "We don't want to become a low-wage tax haven bargain basement economy off shores of Europe."
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