Share the Google Photos to Instagram using the iPhone

Sorry but you don't really have any control over this. Some sharing panels offer a "Open In..." option in the bottom-right corner. If you check an application that does offer this option (for example Dropbox) you'll likely find Instagram in that list.

For some reason Apple doesn't provide that "Open In..." option in the Camera or Photos apps.

The Facebook and Flickr options are hard-coded by Apple. They always appear in the sharing list even if you don't have the applications installed.

As Andrew Ferrier stated in the comments to the question, this is indeed a part of Facebook's privacy settings. Here is how you set it to share to "Friends" by default:

1. Go to your Facebook.com.
2. Find the page to edit your Instagram settings.
   • this should be on YourLinkToTimelinePage/app_instapp
   • or you should be able to find it at https://www.facebook.com/settings?tab=applications
   • or in the timeline menu under your profile picture, find it in More

3. Change your App visibility and post audience setting to Friends

   This setting controls who on Facebook can see that you use this app. It also allows you to choose the audience for posts the app makes on your behalf.

Hope this helps you.