Automatically saving gmail messages to icloud

There is no way to dynamically use iCloud to store your Google email. You have a few options if your Gmail is full:

• Before undertaking any major data cleanup, I would recommend backing up your Google data: https://support.google.com/accounts/answer/3024190?sjid=5667277158613297892-NA
• Clean up your Google storage. In particular search for large items such as videos that you no longer need and delete them from your storage (https://support.google.com/mail/answer/9312312?hl=en)
• Apple does offer a method of importing email from another provider into iCloud email (Import emails from another account into iCloud Mail - Apple Support). However, understand that that would simply do a one-time import of those emails. All new emails would continue to only show up in Gmail. This therefore doesn't solve your problem unless you deleted a pile of old emails from Gmail after confirming a successful import to iCloud for archival purposes.