While media watchdog Ofcom is tasked with reviewing the channel's output, Channel 4 is state-owned and its existence is underpinned by legislation that could be changed by parliament.
Shadow culture secretary Tom Watson wrote to Ofcom about the matter and said it was "deeply concerning that a governing party would wish to restrict the free press".
Co-leader Patrick Harvie, speaking at a demonstration in George Square, Glasgow, on Friday, criticised Mr Johnson after he opted out of the debate hosted by Channel 4 News on Thursday evening.
"It is not for Government to seek to impose its will on an independent regulator".
Justifying Channel Four's move, its news editor Ben de Pear said: "These two ice sculptures represent the emergency on planet earth, not in any human form but are a visual metaphor for the Conservative and Brexit parties after their leaders declined our repeated invitations to attend tonight's vital climate debate".
In his letter, he said: "I'm urging you to call out this meddling and demand that whichever political party wins the next general election allows Ofcom to operate free from political interference".
Former Tory culture secretary Ed Vaizey added to the criticism, saying it is "not a sensible strategy for political parties to threaten broadcasters".
"However, as we made clear to him repeatedly, because he is not the leader of the Conservative Party, his participation was not required".
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A Conservative voter asked the Conservative leader: "What can you say to me today when you can't even debate climate change last night".
Rebecca Newsom, head of politics at Greenpeace UK, said: "Given how poorly Boris Johnson's manifesto scored in our climate and nature ranking, it's no surprise he refused to take part".
On November 22, a peak audience of 4.6 million watched Mr Johnson, Mr Corbyn, Liberal Democrat leader Jo Swinson and SNP leader Nicola Sturgeon battle a barrage of hostile questioning on a BBC Question Time special. If you value what we are doing, you can help us by making a contribution to the cost of our journalism. Unlike in the US, television broadcasters in Britain face strict rules on being politically impartial during election periods and can face fines if they do not comply.
Leaders of the major political parties from around the United Kingdom were present for the debate, except for Johnson and Brexit Party leader Nigel Farage, who were instead represented by frozen sculptures marked with their party name.
Mr Corbyn was interviewed by BBC's Andrew Neil this week, with many believing the interview went disastrously for the Labour leader.
Mr McDonnell added that if Mr Johnson does agree to take part, he would have "played" the BBC by pushing his appearance beyond the point when many postal ballots are expected to have already been returned.
"And so what he's doing now is he's avoiding, he's running scared".