Streeting rejected her apology, saying she was “only sorry because she was caught and called out and said the quiet bit out loud”.
Asked about Pochin’s comments, Streeting said: “What she said was a disgrace, I think it was racist.
“What we have seen is a return of 1970s and 1980s-style racism I thought we had left in the history books.
“The only way we are going to defeat this racism is to call it out and confront it for what it is.”
He accused Reform of failing to speak for the whole country, saying: “They think our flag only belongs to some of us who look like me, not all of us who built this country and built its success.”
Streeting also condemned Nigel Farage’s lack of public comment about Pochin, accusing him of “deafening silence”.
Pochin was previously criticised by then party chairman Zia Yusuf for suggesting a ban on the burka during a speech in Parliament.
Responding to her comments on Sunday, Yusuf – who is now the party’s head of policy – said Pochin was “right to apologise” but said she was raising “a very valid point we must be able to talk about”.
Speaking on Sky News, he said Pochin was “a lovely person” and her point had been “statistically borne out by a Channel 4 study”.
Asked about her comments, Conservative shadow home secretary Chris Philp said he would not “call her racist” and said “there are legitimate concerns the public have about mass migration that need to be addressed”.
“It’s certainly not language that I would use and I don’t think politicians should speak in those terms,” he added.
The Liberal Democrats have called on Farage to withdraw the whip from Pochin over her comments and called criticised Philp for “refusing to call out blatant racism for what it is”.
Party spokesman Max Wilkinson said: “Nigel Farage keeps insisting that racism has no place in his party.
“Now is his chance to prove it – he must withdraw the whip or concede that Reform tolerates blatant racism.”