The local infrastructure body has said it will not support the pledge and will instead sign up to an alternative Local Government Association pledge, which is focused on small and medium enterprises and the voluntary sector.
Navca is calling on other voluntary organisations to back its position.
A statement from the organisation said the government procurement pledge "takes a top-down approach that places too much emphasis on multi-million pound companies to the detriment of the voluntary sector and local businesses".
It said that, in contrast, the LGA’s draft pledge, which is currently out for consultation, "starts with users and citizens and gives a greater emphasis to voluntary organisations".
Joe Irvin, chief executive of Navca, said: "The government’s pledge is a good idea that has been executed badly. The LGA’s pledge is much more voluntary sector-friendly which is why Navca is happy to support it."
He said the LGA had also recognised that there was currently a risk-averse approach to public sector procurement and that a change in local government culture was needed to address this.
Ralph Michell, director of policy at Acevo, said the chief executives body had signed up to the Cabinet Office pledge. He said the pledge was "a step in the right direction," but that there was still much more the government could do to improve procurement, such as developing a commissioning academy that would help develop the skills needed to commission from charities and social enterprises.
"We’ll be pushing them to get that work done," he said.
A Cabinet Office spokesman said: "Since May 2010, the amount of government business going to SMEs is on track to have doubled. By increasing transparency and cutting bureaucracy, our reforms are helping businesses of all sizes and shapes.
"We are pleased that more and more organisations, including Acevo and Navca, are pledging their support for better and simpler procurement. Whether they sign the central or local government pledge isn’t the issue – what matters is that all types of providers see the benefits of better public sector procurement."