Bill Gates last week posted a three-second video on Instagram of himself hanging a sign in a window reading “Thank you health care workers.” Over the following days, the post was pelted with hundreds of thousands of comments painting Mr. Gates as partisan, accusing him of crimes against humanity and linking him to various conspiracy theories involving vaccines, the World Health Organization and implanted microchips.

As one of the richest men on earth and a vocal public-health advocate, Mr. Gates has long been a target for...