He’s making me feel like a horrible mother

Yesterday was a particularly difficult day, baby refused to sleep and was very fussy. I think her tummy was bothering her. By the time my partner got home from work, I was at my wits end. Extremely frustrated and tired. I sort of broke down and told him how difficult my day was and he proceeds to make a comment about how it must have been a hard day if I didn’t get my usual 12 hours of sleep in (because he knows I sometimes nap with baby) I walked away to shower, rushed my shower because I felt guilty even doing that for myself. I tried to talk about it, and all he had to say was “If you want out, tell me. Nobody needs you here.” Also offered to hire a babysitter. I explained to him I don’t need that, I’m not even sure why he would say that. I just had a hard day because I could hardly get dinner and all my chores done, and once I said that he said “I never asked you to do any of that, I’ll do it myself”. We continue to argue about how he’s not giving me the support I need and why he would even make the comments that he did, and he then took it even farther by saying he thinks I have it in me to walk away from my child. Of course this cuts deep because I would never. Our entire conversation made me feel like a horrible mother, and on top of it he was able to get her to sleep after we gave her some medicine for her tummy, and because of that he tells of I had just paid a little attention to her I would have figured out that’s all she needed. I said he was making me feel guilty and like a bad mom and all he said was that it’s not his fault I feel that way, it’s mine. I’m so hurt and feel like I’m already failing and not even getting his respect as the mother of his child.