Mr Corbyn insisted he was carrying on with the responsibility given to him when elected leader by an overwhelming majority of members and called for the party to “come together”.
He dismissed claims he half-heartedly campaigned for EU membership and so contributed to the vote for Brexit, pointing out that two thirds of Labour voters backed Remain.
The Opposition leader said his party has won every by-election it had stood in since he became leader and increased majorities in some seats.
And he took credit for forcing the Tories into U-turns on cuts to tax credits, disability benefits and police funding, along with the forced academisation of schools.
His comments are likely to infuriate many Labour MPs who voted 172-40 to back a motion of no confidence in the leader. Mr Corbyn is still in post a week after the vast majority of his shadow cabinet and frontbench team resigned in an effort to force him out.
In a defiant video posted on Twitter, Mr Corbyn called for unity.
The Labour leader said: “After the events of the past week, I wanted to talk directly to Labour Party members.
“Only nine months ago I was very honoured to be elected leader of our party with 60 per cent of the votes, I have a huge responsibility, I’m carrying out that responsibility.”
He spoke after Angela Eagle delivered a clear threat to Mr Corbyn that she will mount a leadership challenge if he does not step down.
Ms Eagle, who was expected to launch her bid for the top job last week, warned that the threat of a contest had not gone away.
She said: “There are many people, MPs, party members up and down the country, asking me to resolve the impasse and I will if something isn’t done soon.”
She added: “I have the support to run and resolve this impasse and I will do so if Jeremy doesn’t take action soon.”