It seems that you have already taken the necessary steps to connect to MySQL server from GCE your instance. Based on the Stackoverflow case here the recommendation to resolve this problem is to comment this line below (add # at the beginning of the line) in your my.cnf file:
skip-networking
Once done you need to restart MySQL service
sudo service mysql restart
On the GCP side, if you have allowed the IP ranges you want to use with the specified port(3306), it should work fine.
You can also use “nmap” command to verify if port 3306 is open on your GCE instance with MySQL installed.
GCP provides CloudSQL which is a managed MySQL instance. You can access CloudSQL from your GCE VM using private address. This feature reached beta recently. This link provides detailed information about using private IP to connect to your Cloud SQL instances.
Before configuring a Cloud SQL instance to use private IP, you need some steps to be taken. This document provides step by step instructions for configuring an instance to use private IP.