NEW DELHI--Bharti Airtel Ltd. on Monday reversed its controversial decision to start charging customers for phone calls made using free messaging services such as Skype and Viber apps on cellphones and tablets.

On Friday, New Delhi-based Bharti said it would introduce special subscription packs for people who wanted to use the internet-based services. That decision started a discussion about “net neutrality” -- the principle that all Internet traffic should be treated equally. India is the world’s second-largest cellphone market...