One user said he had been given access to around 50 videos that did not belong to him under the Consultation Replay section of the app, as he checked a prescription.
Babylon said it had fixed the issues and notified the Information Commissioner.
The app, which has more than 2.3 million users in the UK, offers live-streamed consultations over smartphones and electronic prescriptions. It also offers a chatbot that triages patients based on their symptoms.
"You don't expect to see anything like that when you're using a trusted app. It's shocking to see such a monumental error has been made," Rory Glover told the BBC.
