Yes. You can keep your registry private.

Desktop- To make your registry Private hover over SETTINGS and click Visitor Setup. Then scroll down to GIFT LIST PRIVACY and select your preferred selection. 

 Mobile App- To make your registry PRIVATE :

- Click on the Menu located at the bottom right of your screen.

- Scroll down to Account Settings.

- Under PRIVACY click the toggle button Allow my registry to be found on MyRegistry.com and/or Allow my registry to be searchable on search engines to disable.