First, the roles:
It's the old old story that children and home are women's work, earning the money is men's work.
Yet you are doing both.
He isn't.
You have a full time job.
He has a full time job
You have another full time job taking care of family and home
He has a part time / casual job 'helping out' with family and home.
That is not a fair division of labour.
Then the fact that you do it more than he does means you're more practiced, skilled and capable than he is. He's not incapable, but he's literally less capable. I think your point was he was playing the 'I'm incapable so I shouldn't have to do this' card.
He needs your guidance both in how to do the essential family work, and how he thinks and talks about his role. It's nice to think 'I'm helping,' but he isn't 'helping', he's co-parenting and needs to get better at it.
Second, the words:
The roles situation caused the words incident.
It seems what you said came out wrong - but you still chose to call him a belittling, insulting name, even though that's one of your no-nos for yourself, and that offended him.
He has an immature way of dealing with that, and lashed out with a misogynistic insult.
He lacks the ability to accept an apology, or to accept his part and apologise to you.
He also fails/choses not to listen and understand you.
None of this is going to be a positive environment for your children.
This needs a calm, considered adult conversation. You both screwed up and it started with falling into the unfair division of labour trap.