Jeremy Corbyn hijacked an appearance on Facebook by Theresa May to issue a direct challenge to the Prime Minister to take part in a TV debate with him.
But Mrs May dismissed his call, saying that it was more important for her to take questions directly from voters.
Mrs May became the first serving leader of a UK political party to take part in a Facebook Live broadcast, hosted by ITV News as part of the campaign for the June 8 general election, answering questions sent in by users of the social media website.
She discussed her experience of type one diabetes, revealing that she injects herself with insulin four or five times a day, and urged fellow sufferers not to allow the illness to hold them back from doing what they want in life.
And she discussed her taste in fashion, saying that her message to other women wondering what to wear in the workplace was: "Don't be afraid to be yourself."
Presenter Robert Peston then told her he had received a query from "Jeremy Corbyn of Islington", who said: "Hello Theresa May, as Prime Minister you've served your elite friends by giving them tax cuts when wages have stagnated, house-building is at its lowest since the 1920s, there are 20,000 fewer police on our streets since 2010 and the NHS is in crisis.
"Do you not think the British people deserve to see me and you debate live and on television?"
Mrs May responded: "What I think is more important is actually that I and he take questions directly from the voters. I don't think people get much out of seeing politicians having a go at each other, I think people want to hear directly."
ITV will on Thursday host a live televised election debate in Salford, and has said that an invitation to the leaders of the seven biggest parties will remain open until the broadcast starts.
But Mrs May has insisted she will not take part in any televised head-to-head clashes with her rivals, and Mr Corbyn has indicated he will not take part if the Prime Minister is absent.
Mr Corbyn responded on Twitter: "Thanks @Peston for reading my question to @theresa_may asking why she won't debate me. It's weak leadership to hide from your record."
Mrs May was also confronted by a questioner demanding to know why former Ukip leader Nigel Farage had not been given a knighthood. The PM laughed but did not reply.
The Prime Minister's Facebook appearance came hours after she was confronted during a walkabout in Oxfordshire by a woman with learning disabilities who complained that cuts in her benefits had left her without enough money to live on.
Cathy Mohan told the PM that she had only £100 a month in Personal Independence Payments to live on and pleaded for the return of the Disability Living Allowance, which was replaced by PIP under Government reforms.
Mrs May told Peston the PIP reforms were "part of trying to ensure that we focus payments on those who most need it, those who are most vulnerable".
She added: "There are a number of issue people raise around PIPs. One is about the assessment process and we have been making changes in that to make that a better process for people. We want to try to help those disabled people able to get into the workplace to do so."
Asked by a fellow diabetes patient about her own experience of the illness, Mrs May said: "I am a type one diabetic. That means when I eat, I have to inject insulin, which I do.
"I will be injecting myself four or five times a day... You just get into a routine. You depend on that insulin and you just build that routine into your daily life.
"The crucial thing to me is being a diabetic doesn't stop you from doing anything."
Mrs May said that she did not mind comments about her fashion sense, repeating a story she has told before about a woman who told her that her distinctive shoes had inspired her to go into politics.
"What I would hope for women out there, be it getting into politics or in business or the workplace generally, is be yourself," she said.
"Obviously there is some employment where you have to wear a particular uniform, but generally don't be afraid to be yourself."
Peston told the PM almost 40,000 questions had been submitted during the 45-minute session, most of which he was unable to read for lack of time.
A stream of comments and questions scrolled past the screen showing the Prime Minister, alongside counters which suggested that around 400,000 people logged on for at least part of the event, with up to 15,000 people watching at any particular point.
Users fired off 9,900 "angry face" emojis during the webcast, compared with 4,300 "thumbs-ups" and 1,200 heart-shaped "likes".
Among commenters was Donna Palfrey, who accused the PM of robotically repeating her slogans of "strong and stable" and "coalition of chaos", and asked: "Why are you scared of a TV debate with REAL questions by REAL people?"
Paul Kelly asked: "Can I ask a question. Is Mrs May happy that her government has put millions into poverty? Is it right that a working person like nurses is forced to use foodbanks?"
Irene Barbara Johnstone voiced her support for Mrs May to be in place to conduct Brexit negotiations with the EU, saying: "If this Prime Minister can't get the job done, seeing Brexit through in our (UK) favour nobody else can!"
Meanwhile Mrs May has reaffirmed her support for a free vote in Parliament on restoring fox hunting, although she said she had never participated herself.
"I have never been fox hunting," she said.
"My view is that it should be a free vote for parliament so members of parliament individually should be able to exercise their view on this matter."
The Prime Minister disclosed she had been the subject of "fake news" claims circulating on social media during last year's Conservative leadership contest to succeed David Cameron.
"During the Conservative Party leadership campaign, we started to see some pretty nasty videos being sent round about me," she said.
"I didn't actually see any of them but I'm told it was in the realms of claims that weren't accurate. So it is a concern.
"But social media does bring huge benefits as well."
Mrs May confirmed if she was re-elected on June 8 she would serve a full term as Prime Minister and would see through the Brexit negotiations, which are due to conclude within the next two years.
"If I am elected I will certainly serve my full term.
"I am pretty certain it (Brexit) can be done in those two years.
"A new parliament will take us through to 2022 which is three years beyond the 2019 and I will be round."