Its publishing director, Stephen Quinn, said he decided not to use the Co-operative Bank's advertisement because it was "tediously controversial" and readers would not like it.
The bank hit back, claiming that Vogue was guilty of censorship and was refusing to carry the advertisement because it was afraid of upsetting cosmetics firms.
The advertisement shows a woman applying face cream and rabbits in a laboratory, and has appeared in poster form around the country.
It is aimed at highlighting the fact that the bank does not invest in companies that test their products on animals.
The row comes months after the magazine was involved in another dispute over advertising. In May, the watch company Omega threatened to suspend advertising because the magazine featured "anorexic-looking" models.
The Co-operative Bank's managing director, Terry Thomas, said: "The only possible explanation for this censorship is that Vogue is frightened of upsetting the powerful cosmetics lobby. This issue of cruelly testing cosmetics on innocent animals will not go away." Mr Thomas said the bank's two million customers were "right behind us". Other glossy magazines would be used in the next few weeks.
Mr Quinn said he had decided the advertisement was not appropriate because Vogue was a fashion magazine and did not publish controversial advertising The advertisement would "play badly with the readership" and did not clearly convey the benefits of banking with the Co-op, he added.
"We don't wish to encourage tediously controversial advertising. It is deliberately suggesting the experiments that go on on animals are in some way morally wrong," he said.
"The Co-operative Bank are not a regular advertiser in Vogue. We are not denying them a privilege they have become accustomed to."
He said he was "not going to be lectured by the Co-operative Bank".