Labour councillor Ewan Aitken had called on the authority to back the proposed rally in Edinburgh on October 22 as “a way in which the voice of those often silenced will now be heard”.
Cllr Dawe said she was unable to endorse a march against proposals by a government that the Liberal Democrats are a part of as “it would be endorsing a march against ourselves”.
She admitted that the impact of the changes to the welfare system could be “devastating” for some and have an adverse effect on some local authorities.
She added: “I have taken every opportunity to make our views known and I have personally lobbied party colleagues at Westminster and Holyrood.”
Cllr Andrew Burns, leader of the Labour group on the council, said: “You have said you have taken every opportunity to pass on views to influence the bill. I believe that, but one way you can do that is to endorse this motion.”