If one co worker has a dispute against another co worker due to he said she said talk, is it ok for them to confront / discuss the issue in a lunch room filled with other co workers even if the dispute is about possible legal actions?

Co worker 1 = the co worker disputing the issue. Co worker 2 is the co worker being disputed and complained too. So co worker 1 heard rumors from other co workers that co worker 2 said co worker 1 grabbed the others rear and co worker 2 reported it to HR. Co worker 1 decides to confront co worker 2 during a sit down inside the lunch room of work. Other co workers are around and present for lunch. Co worker 1 tells co worker 2 that it better not happen again or will call a 1-800 number to make a report against co worker 2. Even without talking to co worker 2 to hear their side of the story or see what might of happened. Is this action ok and or is this something that should have been done with the manager in private?