Three-time Tour champion Pogacar hit the tarmac late in the 156.8km stage around Toulouse after striking the rear wheel of Johannessen.
The Slovenian got back on his bike, and his rivals Jonas Vingegaard and Remco Evenepoel allowed him to catch back up, limiting the damage suffered by Pogacar.
"Luckily, I've just lost some skin," said Pogacar, who is second overall in the race, trailing Irishman Ben Healy by 29 seconds after 11 stages.
Johannessen quickly apologised for the incident but expressed his horror at the "very scary" messages of hatred directed his way.
"I am terribly sorry for what happened to (Pogacar)," Johannessen wrote on X.
"I was trying to follow a move, and I can see that I was too close. I thought everyone would move to the right, but I made a mistake and would like to say sorry again. I hope he is as good as he can be after a crash like that.
"I would of course like to do it again and do it differently, but I can't.
"But I would not wish anyone the amount of threats I get in my inbox. I am terribly sorry, but also terrified of the hate from all the people. This feels very scary."
Frenchman Bryan Coquard came in for similar treatment after he was involved in the crash that knocked Belgium's Jasper Philipsen out of the race on stage three.
Philipsen, who won the opening stage in Lille, was forced to withdraw after suffering a broken collarbone and two fractured ribs, but said he did not blame Coquard.
Catch up on all the action from stage 11 in our report here.