Mr Darby made the remark about Conservative Parliamentary candidate Norsheen Bhatti during a debate last night in Bentilee.
Responding to a resident's question, Miss Bhatti commented "there is nothing British about the BNP".
But Mr Darby cut in, saying: "I'm more British than you are."
After she demanded an explanation, Mr Darby told the Birmingham-born candidate: "You made a personal attack on me and you got one back."
At the beginning of the debate, Miss Bhatti had complained to organisers after discovering that she had been sat next to Mr Darby.
Miss Bhatti eventually moved away from the far-right politician, leaving a conspicuously empty seat between them.
Another candidate, Liberal Democrat John Redfern, refused to attend because he did not wish to share a platform with Mr Darby.
Audience member Dawn Kelly, aged 40, who lives in Bentilee, said afterwards: "What Mr Darby said sounded racist and there was no need for it."
Mr Darby and Miss Bhatti are among 10 candidates contesting the Stoke-on-Trent Central Parliamentary seat but only half attended the event.
They all pledged to fight to keep a high school in the Bucknall area under reorganisation plans.
The debate was organised by the Community School Action Group, which is battling to keep a school on the site of the Mitchell Business and Enterprise College, in Bucknall.
Stoke-on-Trent City Council plans to merge the school with Edensor Technology College, in Longton, to create a new academy outside Bucknall.
BNP Mr Darby said he was opposed to forcing white and Muslim pupils to mix at an academy he called "a huge social engineering factory".
Independent Gary Elsby said: "The new school should be built on the Mitchell site."
Miss Bhatti blamed Labour leaders for not listening to residents' wishes, and said: "Rather than telling you what the solution should be, we want you to tell us."
Matt Wright, of the Trade Union and Socialist Coalition, said: "Our solution is to refurbish the existing community schools and use the money we save to recruit more teachers to reduce class sizes."
Independent Brian Ward told campaigners: "I wouldn't come here and ask for your vote if I wasn't prepared to back you."
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