The former prime minister unveiled his mission to persuade the UK to stay in the EU, saying Brexiteers voted “without knowledge” on June 23 last year.
Mr Blair said Brexiteers had benefited from a “propensity for revolt” but their views were not set in stone.
He was then asked by BBC political reporter Carole Walker whether he was the right man to deliver such a strong anti-Brexit message.
Replying Mr Blair said: “You can like the messenger or not like the messenger and this is a free country.
“I have got a right to speak and you have got the freedom to listen or not.
Tony Blair grew tetchy after being quizzed over his anti-Brexit speech
“You don’t want to listen to me don’t listen to me.
“I am very clear and simple on this I know there will be a volume of abuse that will come my way for speaking but I am speaking because I believe in it and I care about the country.”
Speaking at the headquarters of Bloomberg, where David Cameron first announced there would be an in/out vote on EU membership, Mr Blair said the referendum result had not provided a mandate for “Brexit at any cost”.
He said the determination of Theresa May and her Government to pursue a ‘hard’ Brexit had put the stability of the union at stake.
The former Labour Prime Minister said: “"I accept right now there is no widespread appetite to re-think. But the people voted without knowledge of the true terms of Brexit.
"As these terms become clear, it is their right to change their mind. Our mission is to persuade them to do so," he said.
I have got a right to speak and you have got the freedom to listen or not
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"What was unfortunately only dim in our sight before the referendum is now in plain sight. The road we're going down is not simply hard Brexit. It is Brexit at any cost.
"Our challenge is to expose relentlessly what this cost is, to show how the decision was based on imperfect knowledge which will now become informed knowledge, to calculate in 'easy to understand' ways how proceeding will cause real damage to our country; and to build support for finding a way out from the present rush over the cliff's edge.
"I don't know if we can succeed. But I do know we will suffer a rancorous verdict from future generations if we do not try."
Following Mr Blair’s speech, Nick Clegg said he agreed with “every single word” the former PM had said.
Jeremy Corbyn was also targeted by Mr Blair and said there was an absence of effective opposition.
Blair and Juncker: Together through the years
Wed, January 25, 2017
Former Prime Minister, Tony Blair and President of the European Commission, Jean-Claude Juncker have always shared a friendly relationship over the years.
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EU Commission President Juncker greets former British PM Blair in Brussels
He added:"The debilitation of the Labour Party is the facilitator of Brexit. I hate to say that, but it is true.
"What this means is that we have to build a movement which stretches across party lines; and devise new ways of communication," he said.
"These groups must find ways of concerting strategy and tactics effectively. We should begin to create informal links immediately and then build them into a movement with weight and reach.
"We need to strengthen the hand of the MPs who are with us and let those against know they have serious opposition to Brexit at any cost."