By default all Gravatar information is public and it automatically appears on hovercards enabled websites. But if you don’t want to show up your information on websites you have commented on then you can make your gravatar profile private. Simply Log in to your Gravatar account. At the top of your web browser’s screen, click My Account, then click on View My Profile and from the lower right corner of your profile, click Hide My Profile option.
The option turns your gravatar profile to private which means your gravatar profile can no longer be displayed on hovercards and widgets etc. But it doesn’t affects your Gravatar image and it shows up on the places you have commented until and unless you choose not to use that image. In-case if you want to remove your image too then click on My Account, then click Manage My Gravatars option and delete the images.
If you are a Jetpack plugin user and just want to disable hovercards on your self hosted WordPress site then first login to your admin area dashboard. Navigate to Jetpack -> Jetpack (Module Page), scroll down to Gravatar Hovercards section then click ‘Learn More’ and deactivate the hovercard module. The option deactivates hovercards for all registered & unregistered users on your WordPress site.