This defeat is the latest loss taken by Labour Party in what has been a stunning collapse of the party over the past decade.
Former London Mayor Ken Livingstone also blamed Mr Blair, who won three general elections, for sending the party into "long-term decline".
"Labour have held this seat since the 1930s".
"I was elected to lead this party".
The Labour Party's strength in Parliament is at its lowest level since 1987, with only 230 of the 650 seats in the House of Commons, and is far behind the Conservatives in the polls.
"We are on course to a historic and catastrophic defeat and that will have very serious consequences for all of the communities that we represent", he told the BBC Radio 4 Today programme. He said defiantly Friday that he is "proud" to lead the party and will continue to focus on the future of Britain's financially strained National Health Service as well as the issues of social care and housing.
Perhaps Ukip's biggest failure was not to get Tory voters to come across to the party in significant numbers to stop Corbyn's Labour from winning. Both constituencies, like so many in Britain, have been let down by the political establishment.
'Heartbroken' Ben Simmons to miss rest of National Basketball Association season with injury
The 6f 10in former LSU standout, who was born in Melbourne, was Philadelphia's first No1 overall pick since Allen Iverson in 1996. But there was always the outside chance that it didn't happen because there wasn't complete or full healing.
Labour beat UKIP in one by-election on Thursday in Stoke-on-Trent, a working class and staunch Leave voting area.
Woodcock stopped short of calling for Corbyn's resignation, but said he should remain leader "unless he decides that he thinks it is right to go".
Asked whether he might be the problem, Mr Corbyn said: "No". But our message was not enough to win through in Copeland.
Baroness Smith, Labour's leader in the Lords and a shadow Cabinet member, told the Huffington Post that Mr Corbyn would be "thinking long and hard" about the Copeland result.
"The leader has to reflect official Labour party policy", she said "T he Labour Party has always been a broad coalition, and on nuclear there's always been differently held views".
The leader of the shop-workers' trade union Usdaw, John Hannett, said that the by-election results " raise questions about Labour's electability and the vision it offers to the people of this country".
"Labour needs a clear, credible message that people can believe in and Jeremy Corbyn needs to take responsibility for delivering that sooner rather than later".