Can a resource manager approve a resource assignment to fulfill a project resource request?

Yes.

Resource managers with the Approve Project Resource for Project Assignment privilege can approve a resource assignment for proposed resources using the Submit and Approve button on the Propose Resources for Request page or the nominated resource that the project manager evaluated and requested for approval. You can't submit and approve resources if you selected multiple resources and the number of resources selected exceeds the remaining quantity or you nominated one or more resources to a resource request.