Swinney said Tuesday there's no place for people who send threats to college students for making mistakes in a game, and that those who do are not "real fans."
"Coach [Chad] Morris told me [about the threats] last night," Swinney told reporters. "That's disappointing. I think he deleted his account. It's a shame that young people have to deal with that kind of stuff. Nobody wants to make a bad snap.
"I think most of it was people mad and venting, but it was threatening to Ryan is what I was told. Those aren't fans -- those are people with issues."
Watson fell on the ball at the Florida State 24, and kicker Ammon Lakip missed a 40-yard field goal attempt.
Swinney said he tried to lift Norton's spirits Monday by jokingly telling the team that Lakip hadn't received any hate mail -- at least not yet.
"I told him it might take two or three days for my letter to get there," Swinney told reporters.