Damian Green said it would be "undesirable" for Parliament to try to pass such a law which would be at odds with the UK's "tolerant and mutually respectful society".
Fellow Tory MP Philip Hollobone introduced a Private Member's Bill which would make it illegal for people to cover their faces in public.
More than two thirds of voters back a ban like that approved almost unanimously by French MPs last week, according to a recent opinion poll.
But Mr Green insisted such a move was "very unlikely" to be copied here.
"Telling people what they can and can't wear, if they're just walking down the street, is a rather un-British thing to do," he said. "We're a tolerant and mutually respectful society."
There were occasions when it was important to be able to see someone's face, he said.
"But I think it's very unlikely and it would be undesirable for the British Parliament to try and pass a law dictating what people wore."
Unlike France, the UK was not "aggressively secular", he said, suggesting the proposed ban across the Channel was being brought in to make a point.
Earlier, Mr Hollobone declared he would not meet with burka or niqab-clad women at his Kettering constituency surgeries unless they lift their veils.
"If she said 'no', I would take the view that she could see my face, I could not see hers, I am not able to satisfy myself she is who she says she is.
"I would invite her to communicate with me in a different way, probably in the form of a letter," he added.
Shadow Justice Secretary Jack Straw, who sparked controversy in 2006 when he revealed he asked constituents to lift their veils, said he, too, was opposed to a ban.
"I was seeking to generate a debate within a framework of freedom," he said.
Mr Hollobone previously described the burka as "offensive" and "against the British way of life".