I work in a startup Software company in India. I head the development team. Recently, "Miss Sue" made some mistake. When "Miss Kate," the Lead of some other team, asked Sue about it, she replied rudely.
When this was brought to my attention, I confronted Sue, she accepted her mistake, and she and I discussed ways in which we can avoid the mistake in future. For me, the matter was over. But Ms. Kate called up a meeting of some senior developers, her own team and Sue, and of course me.
During the meeting, although the issue was resolved, Kate verbally attacked Sue with the purpose of further humiliating her. I defended Sue and asked everyone to focus on the solution for the problem, rather than on who made what mistake. When we were done discussing solutions, I asked everyone to leave. This angered Kate and now she has completely stopped talking to me.
I would like to also note that Kate kind of has a reputation of talking with people in a higher pitch/humiliating them in front of others, and in her own words, she thinks, 'That is how people learn'. I have always opposed that way of handling mistakes, and have even tried to explain it to her. On all previous occasions, she accepted she was wrong, and said she won't repeat it, but then again she keeps doing this kind of thing.
That is the reason why I have resisted explaining same things to her. I want her to realise her behavior would have propagated negativity within the team. But at the same time, this not talking thing is making me uncomfortable.
For last few days I have been thinking of raising the issue to our superior, who happens to be the CEO of the company. But, I am unsure of doing this, because I am not sure if this situation is worthy of taking up to the CEO. Can you please help me with clearing out my confusion?