Is it possible for a very large scale service to use a major public cloud managed database service?
For example, a service with a scale of Netflix and a service like Amazon Aurora.
If not, which solutions are available for very larges scale services?
Yes, it is certainly possible, but it entirely depends on the architectural design of the entire system. Separating application services into manageable components (i.e. microservices) is one approach to this. While Netflix as a whole is one application in terms of the user experience, it is comprised of many applications/services that provide a small fraction of the functionality needed to run Netflix as a whole.
In terms of a database, the same principal can be applied. Netflix doesn't manage one database for the entire application, each component utilizes it's own scalable database service.
My conclusion after additional research and Preston Martin answer:
If the workloads doesn't require more resources than the maximum supported by the service:
maximum horizonal and vertical (instance type) scale of the servers.
If it does, the data should be distributed across multiple managed database clusters.
If that doesn't meet the requirements, this solution should help: