Ken Livingstone has "unreservedly" apologised for suggesting a shadow defence minister who criticised him needed "psychiatric help".
Kevan Jones - who has suffered with depression - said the comments were "gravely offensive".
Mr Livingstone initially refused to apologise despite being urged to do so by Labour leader Jeremy Corbyn, who had just given him a new policy role.
But he has now tweeted that the comments should not have been made.
The decision by Labour's National Executive Committee, chaired by Jeremy Corbyn, to appoint the former London mayor as joint chair of Labour's defence review sparked an angry backlash from pro-Trident Labour MPs, with shadow defence secretary Maria Eagle reported to be "furious".
Ms Eagle, who was leading the review, said she was not consulted before Mr Livingstone was made her co-chair - the pair hold opposing views on Britain's nuclear weapons.
Junior defence minister Kevan Jones questioned whether Mr Livingstone knew anything about defence, prompting the former mayor to tell the Daily Mirror Mr Jones was "obviously very depressed and disturbed" and "should see a GP".
Mr Livingstone tweeted his apology after refusing to say sorry during interviews.
"I unreservedly apologise to Kevan Jones for my comments. They should not have been made at all, let alone in this context," he said.
"I also make this apology because Jeremy [Corbyn] is right to insist on a more civil politics and as a party we should take this seriously".
Earlier he told the BBC he "had no idea" about Mr Jones's mental health condition.
Asked whether this was an excuse for his language, he told BBC News: "I grew up in South London where if someone's rude to you, you are rude back."
He said Mr Jones should first withdraw his criticism, "and then I'll go for a drink with him", adding that Mr Jones had "undermined the leadership" with his remarks.
In 2012, Mr Jones was praised after speaking in Parliament about his mental health battles and the "difficult" decision to go public.
Mr Corbyn's spokesman reacted to Mr Livingstore's comments by saying the Labour leader was "incredibly concerned that people with mental health problems shouldn't be stigmatised".
He added: "He has worked with Kevan in the past on this issue and is impressed by his bravery in speaking out on his own mental health issues. Ken should apologise to him straight away."
Former shadow chancellor Chris Leslie called on Mr Livingstone to resign, and shadow mental health minister Luciana Berger said "any reasonable person should be appalled" at what he had said.
Shadow foreign secretary Hilary Benn said the comments were "very unfortunate" and it was right he had apologised.
Conservative MP Charles Walker, who went public with his own mental health difficulties at the same time as Mr Jones, said he was "absolutely seething" about Mr Livingstone's "vile" remarks.
The clash came after Mr Livingstone's appointment highlighted divisions within the party over the future of the UK's nuclear weapons system.
Ms Eagle believes Trident should be renewed, while Mr Livingstone - who declared London a "nuclear free" zone in the 1980s when he was in charge of the GLC - is a longstanding critic and would like it to be scrapped.
BBC political correspondent Carole Walker said Ms Eagle was not considering resigning but was "fuming" that as the first female shadow defence secretary a man had now been appointed to "look after" her.
In a BBC interview, Ms Eagle said she was "sanguine" about working alongside Mr Livingstone and had found out about his appointment on social media.
Earlier Mr Jones, who has held the defence brief for seven years, told PoliticsHome the appointment would damage Labour's "credibility" on defence.
He said: "I'm not sure Ken knows anything about defence.
"It will only damage our credibility amongst those that do and who care about defence."
Labour said Mr Livingstone was now a "co-convener" on its policy commission.
A party source said Ms Eagle and Mr Livingstone - the former Labour MP for Brent East - would oversee the policy review, but stressed Labour policy would be decided by its National Policy Forum.
A £20bn like-for-like replacement of Trident was agreed by the last Labour government and its renewal has been supported by the party since.
But Mr Corbyn, who was elected Labour leader on anti-Trident platform, said the party's official position on its renewal will form part of the defence review.
The government is due to decide whether to renew Trident in 2016.
Ms Eagle - who criticised Mr Corbyn for saying he would not fire Britain's nuclear weapons if he were prime minister - has previously said she will attempt to persuade the leader to take her stance in favour of renewal.
The SNP - which wants to scrap Trident - is understood to be planning to stage a debate on Tuesday.
BBC assistant political editor Norman Smith said Labour MPs were being instructed to abstain on the vote - but some were planning to defy Mr Corbyn and vote against the SNP motion.
Their argument will be that as existing Labour Party policy is to support the renewal of Trident, they should therefore be able to vote against the SNP's motion to scrap it.