Question
It is possible to access the Internet from a private subnet only with NAT without an Internet Gateway associate with its VPC?
Background
VPC with public and private subnets (NAT) says below. Just with this sentence, it would be a fair conclusion that it only requires NAT to reach the Internet without an Internet Gateway attached its VPC.
A NAT gateway with its own Elastic IPv4 address. Instances in the private subnet can send requests to the internet through the NAT gateway over IPv4 (for example, for software updates).
I believe for any traffic from a VPC to reach the Internet, the VPC requires the Internet Gateway but the document does not clearly say so, hence being confused.
The NAT Gateway needs Internet Gateway to reach the internet.
Technically the hosts in the private subnet don't need IGW to reach out but the NAT GW does, so ... you can't really do it without IGW. Unless you do some creative routing with multiple VPCs, some with and some without NAT/IGW. Perhaps using Transit Gateway? But that's a whole new level of complexity.