On the face of it, I don’t think either of you are wrong. It sounds to me you’re both exhausted. It’s likely he’s struggling with the lack of sleep and crushing responsibility of having a child. When people threaten to walk away and ask for reassurance, it’s usually because they’re in need of love/a hug.
Do you know why he said he doesn’t like how you speak to him? Have you snapped at him? Or been criticising him? It’s a common response when people are exhausted and need some TLC. And you must be shattered.
It’s common for couples to turn on each other, rather than giving each other the love and care they need. It’s because you both need love and care, so can struggle to give it to each other - as well as a baby.
When one or both of you starts to get like that, try to hug it out, find the humour in your situation and view yourselves as a strong team, who can get through things together. Try to remember what made you so close before you had the baby.
I really recommend you try to have a little break if you can, depending on the baby’s age and what support you have. A nice lunch and walk (rather than dinner when you may be tired), a massage together, a night in a nearby hotel. Even if you take the baby, the change of scene may help. Make it a special time. But try to have a little talk about how you both respond under stress, how that makes the other feel and what you both need to work on.
For example, he shouldn’t threaten to leave you when you’re tired and dealing with a crying baby. That would likely make someone feel vulnerable and rejected at a vulnerable time, and can be controlling and manipulative. Instead, he should say: ‘I’m struggling to cope right now and I need some reassurance - I bet you do too”.
And certainly keep an eye on it and say up-front if you find his behaviour isn’t improving and a pattern is forming. Being emotional, needy and dramatic at times of stress is pretty normal. But if it keeps happening and he won’t work on it, then that’s a problem.