Can private fargate instances access public sites via internet gateway instead of a NAT?

Once the private subnet have a default route to the Internet Gateway, the subnet will become a Public subnet under AWS terminology for Subnets. The other piece of this is that your container should be assigned with a Public IP for its traffic to route the internet. Thus you either assign a public IP to your containers to access the Internet directly behind the Internet Gateway or use a NAT gateway to do the translation behind the scene (Private IP <> Public IP) in case your containers are only assigned with Private IP.