If I know the exact name and date when someone died, is there any way to find out their cause of death?
Some locations have public record copies of death certificates (issued without the social security number), but I don’t know how widespread that is. Generally, only the family can get a death certificate, and only through the mortuary that handles the remains; but it does contain the doctor’s estimate of the cause of death, assuming no autopsy (i.e. no questionable circumstances in the death, nor any association with criminal activity).
Otherwise, you can check the funeral notice (Legacy.com is a good source in most locations) but if the family did not choose to share the information there, you are probably out of luck. In general, cause of death is not considered anyone else’s business.
You could kill them yourself. Then you’d know the cause of death.
If they’re a U.S citizen, you’ll probably find them on the Social Security Death Index.
This does not appear to list cause or manner of death.
Here, at any rate, death certificates are public records and anyone may obtain a copy (with cause of death).