Is there a way to disable the ability to delete messages on iPhone

You cannot install a restriction to prevent deletion of Text Messages on the iPhone.

Furthermore, You can suggest this feature directly to Apple by leaving a product feedback here 👉Product Feedback - Apple 

Axel F.

Addat Said:

"Is there a way to disable the ability to delete text messages, ie. In the case of a teenager deleting messages so that the parent can’t see them?"

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Disabling Deletion of Messages:

Disable the payed-for service of Messaging all together. Messaging is a service you pay for through your Carrier. So, cancelling that service would avoid all of this.