Who could object to a movie star visiting a refugee camp? Yet when Priyanka Chopra visited Rohingya refugees in Bangladesh last week, Indians bombarded her Instagram and Facebook accounts with protests. Some demanded to know why Ms. Chopra, best known in the U.S. for the TV drama “Quantico,” had not visited Hindu refugees who fled Muslim-majority Indian Kashmir nearly three decades ago.

India’s response to the Rohingya crisis next door has been hijacked by domestic identity politics. Though India borders both Myanmar and Bangladesh,...