Why won't Apple allow an app like Covenant Eyes to monitor internet activity on the Safari browser on iOS?

This is a user-to-user forum. No one here would know what Apple's reasons are unless they have publicly announced them. Speculation on Apples policies is prohibited by the Terms of Use of these forums. You can submit feedback on features you'd like using the appropriate link on the Feedback page:

http://www.apple.com/feedback

Are we talking about this program in the iTunes store, which is freely available for iPhone and iPad??

http://itunes.apple.com/us/app/covenant-eyes/id335318146?mt=8

Apps run in what's called a sandbox. They cannot interact with other apps except to send a data file from one app to another (like PDF) if the sending app supports that.

In the case of your app, it cannot interfere with any other app. Example: When you have PDF file A on app X and "open ..." it in app Y, a copy of the file is sent from X to Y. App Y cannot reach into X's space and grab A . This prevents malicious software from causing havoc.

As covenant explains on their web site and in the iTunes store app description, the inherent design of the iPhone and iOS disallow any app like that from interfering with the operation of another app. But you can do as they suggest - use restrictions to disable Safari and disallow the installation of new apps. That leaves their own app/browser as the only means of surfing the web, and therefore Covenant's filters and monitoring apply.