Journal published a paper, ignoring my objections as a referee
EDIT:  The question has been changed since this was written.

Your key misconception is that the editor needs reviewers' permission to publish a paper.  Actually, the decision to publish rests solely with the editor.  

In this case, you disagree with the editor, but we do not have enough information to tell who is correct.  If you think the errors in the paper are important, then once the paper is published you may be able to submit a comment to the journal.  Do not do that before the paper is published, because for most journals you must maintain the confidentiality of peer review.