Buddhism says that people who look at the sun without stabbing their eyes will die, right?

Orthodox Buddhism does not tell fortune or predict anything. When the Buddha was alive, he made a commandment not to allow later monks and believers to participate in fortune-telling and other activities.

Don't believe those superstitious remarks, but read more formal Buddhist classics, such as Shurangama Sutra, Diamond Sutra, Heart Sutra, Goblet of Fire and so on. To take a step back, US President Trump saw the solar eclipse with the naked eye, and he didn't respond at all. He has lived to this day.