My first thought is to say that your husband is using his cultural and religious differences as an excuse to get out of giving you much needed help. There are no rules within either his culture or religion that say he should not help you to care for your child. In fact, it has been my personal experience that Middle Eastern husbands and wives often work very much together as a team and share many of the responsibilities.
My father-in-law was born and raised in Palestine. After his family had to leave Palestine, he lived in Jordan for many years and then came to the United States in the mid 1960's. He is Muslim like most of the rest of the family. He was also a very involved father and helpful husband. In fact, I have met few husbands that are as caring and helpful as he is to his wife. I recently went through IVF treatments and he was hoping that I would become pregnant with twins. He told me that if I did, he would come here and prop my feet up and cook and clean for me the whole time so that I didn't have to do anything. Now, does that sound like what so many in the western world perceive to be the stereotypical Muslim man?? :)
My father-in-law's mother had seven sons and they all learned to help around the house and take care of their family. All of the brothers that I have met (and the younger generation of sons and cousins) have been very attentive, helpful husbands and fathers around the house. Just as there are still chauvanistic American-born men who believe that's "women's work", there are, of course, Middle Eastern men with a similar attitude. But to blame it solely on the culture or religion is really not accurate. Maybe that's how he was raised within his family though. However, there's nothing saying he can't change! It's time for your husband to reinvent himself and become the husband and father that you and your daughter need. I really think you just need to sit down with him and have a very serious heart to heart. Religion and culture aside, you two are now husband and wife and that transcends everything else. He needs to listen to you and you two need to work through this in order to keep your marriage together.
My husband was raised by a man that was born and raised in the Middle East in a Muslim family and yet he cooks, does the laundry, does the dishes, changes dirty diapers, gets up with the baby on weekend mornings, sweeps the floor, etc., etc. There is absolutely no good reason why your husband can't be more helpful to you, too.