NEW DELHI—Antivaccine misinformation, some of it from social media posts in the West, is spreading in India on WhatsApp, undermining efforts to root out measles and rubella in a country where tens of thousands of people are struck by the diseases each year.

Dozens of schools in Mumbai have refused to allow health officials to carry out vaccinations in recent months, largely because of rumors shared on Facebook Inc.’s popular messaging app about the supposed dangers.

Several...