Does RDS in private subnet inside AWS VPC need a NAT instance/gateway?

Solution 1:

General Rule

NAT provides outgoing internet access, and does not allow incoming connections. Note that NAT gateway is a managed service, and is generally preferred over a NAT instance.

RDS does not need to initiate connections to the internet, so it does not need access to a NAT instance / gateway. RDS is fully managed by AWS, including software updates of database that runs RDS (MySQL, PostgreSQL, etc).

RDS Replication

With database replication internet ingress is required, but that isn't provided by NAT.

RDS Exposed to the Internet

You can expose an RDS instance to the internet (aka "make it publicly accessible"), but that's incoming rather than outgoing so NAT isn't required.