If you want to translate "Sasha gave birth to a girl" literally, it should be something like: "Саша дала рождение девочке." (рождение - direct object, девочке - indirect object.) It is correct grammatically in Russian, but uncommon and the sense is ambiguous.
The literal translation of "Саша родила девочку" is "Sasha procreated a girl". "девочку" is in accusative, it is direct object.
One can say "Саша родила мужу девочку": "Sasha procreated a girl for the husband". "девочку" (accusative) - direct object; "мужу" (dative) - indirect object.
So, when you say "Саша родила девочке" it is "Sasha procreated for a girl", which is incomplete. One should ask, what exactly did she procreated for a girl.
In Russian direct object is usually in accusative. All cases but Acc. and Nom. are indirect cases.