NEW DELHI—India’s Supreme Court on Wednesday ordered tobacco companies to comply with a new government rule requiring that graphic health warnings cover 85% of every cigarette pack, even as the industry pursues a legal battle against it.

Cigarette makers have filed more than two dozen lawsuits against the measure, arguing, among other things, that it infringes their trademark rights. Those lawsuits are continuing. Public-health officials say the new labels will discourage smoking in a country where tobacco-related illnesses...