Johnson, initially diagnosed with an abdominal injury, is expected to be in the hospital for another day for observation, the team said.
Browns coach Kevin Stefanski said Johnson was diving to break up a pass intended for Bryant during a red zone drill.
"[We] would've preferred that he held back instead out of an abundance of caution and not dive for it, but they kind of both dove and Harrison landed on top of him," Stefanski said. "Really unfortunate, unlucky. Hoping for him that he will be back when he is ready. Really the most important thing right now is that he is safe, and we want him to get well very soon."
Stefanski praised the team's medical staff for being "very cautious" with Johnson and recognizing the potential for a serious injury.
The Browns signed Johnson to a one-year deal this offseason, with hopes that he would win the starting nickelback job.
Johnson's is just the latest in a series of injuries the Browns have endured during training camp. Center JC Tretter and linebacker Mack Wilson are both out with knee injuries. Running back Nick Chubb had to enter the concussion protocol after taking a hit on the first day of padded practice earlier this week.