To create the private API you need to already have in your environment : a VPC with an internet gateway, a public subnet with a NAT and an EC2 instance, a Security Group that allows port 443 from anywhere, a VPC Endpoint for execute-api associated with the private subnet, the security group and with Private DNS names ENABLED
To be able to invoke a private API you need to : Log into an instance that is in the same VPC and subnet as your VPC Endpoint and in the same security group or which security group is allowed to make requests to the Enpoint's security group
On the instance, open a terminal and execute the curl command with the API URL