No, a project does not need to be in a "Healthy" status to go into a "Deleted" status. 

A project can go from 'Critical' to 'Deleted' if there has been no cost data in a 32 day retention period. 

There is no way to manually switch this status to "Deleted". Tanzu CloudHealth support does not have the ability to make this change either. The project must pass the 32 day retention period. 

More info on shutting down GCP projects here:

https://cloud.google.com/resource-manager/docs/creating-managing-projects#shutting_down_projects