He told the man, who said three of his grandparents were born outside Britain, that he could class himself as "civically British" but not "indigenous British
His remarks came as he took calls from listeners on BBC Radio 4's The World at One.
During the phone-in, Sean Fowlston from Nottingham asked: "Would you be good enough to tell me whether I am British or not, given that three of my grandparents were foreign-born?"
Mr Griffin said: "It doesn't matter where on earth they come from, obviously I can't you see you down the radio."
He then added: "You're British."
But pressed why it would make a difference what colour the man was, Mr Griffin went on: "It would make a difference in particular, if he was what the BBC would call white, then I would assume from his name he was Irish, and I am part-Irish as well.
"We regard the Irish as completely part of Britain."
Asked why he would need to see him to know whether he was British or not, Mr Griffin said: "Because if I could see him I could tell whether his three ancestors were Irish or not.
Mr Fowlston said: "I was expecting that response."
He then asked: "If three of my grandparents were not ethnically British, would I be British?"
Mr Griffin said: "You would be civically British, yes absolutely."
Mr Fowlston said: "But not indigenous British?"
He was told: "No of course not."
During the phone-in Mr Griffin was asked if the Royal Family were "indigenous" British as their ancestors originally came from Germany in the 18th century.
Mr Griffin said: "Erm, well that was part of the Royal Family. They are a fairly old mixed-up bunch of Europeans, but they are fully integrated.
"Once you reach a stage where you simply can't tell, when the whole community regards someone as being part of the indigenous, that is when they are indigenous.
"That is the case in the Amazon jungle, why is it different here?"
Mr Griffin also defended his election leaflets, many of which feature his image alongside former Prime Minister Winston Churchill.
And he maintained his position that he could not talk about claims he had denied the Holocaust because of "ridiculously strict" European laws.
The BNP is fielding about 330 candidates in the General Election.
In an email to BNP supporters today, Mr Griffin revealed the party's funds were "stretched to the limit".
Seeking to build on Gordon Brown's "bigoted woman" blunder this week, Mr Griffin called for donations to fund newspaper adverts in key areas such as Barking and Dagenham, Stoke-on-Trent, Leicestershire, Manchester and Barnsley.
Asking supporters to "spare a paltry £20", he wrote: "Our activists are already at full stretch and we have no spare finances to fund extra campaign material, so what we want to do is to raise enough extra funds to place newspaper adverts in a whole range of key target areas up and down the country."
He went on: "Our existing funds are already stretched to the limit paying for the huge costs incurred in running a national election campaign.
"If we are going to make this happen, we need to dig deep right now. This could mean the difference between a breakthrough or not in the General Election."
The Independant
