Does a NAT gateway require an Internet gateway?

The simplest answer is YES.

The instances in the private subnet can access the Internet by using a network address translation (NAT) gateway that resides in the public subnet. So, you should have a VPC with both private and public subnets. A NAT gateway must be created in a VPC with an Internet Gateway. Otherwise, the NAT gateway won't work.

For more details: NAT Gateways