A pregnant Woman had intercourse with her husband and directly went into labor. Did she have to perfom ghusl (shower taken in case of major impurity) before she going to the hospital? Is that obligatory?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

Major ritual impurity entails ghusl, while minor ritual impurity entails wudhoo (ablution); however, the person is not obliged to purify himself from the ritual impurity, minor or major, except when he wants to perform an act of worship entailing ritual purity, such as prayer or the like. It is recommended for him/her, though, to hasten to purify himself/herself from ritual impurity. Allaah, The Exalted, says (what means): {O you who have believed, when you rise to (perform) prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles. And if you are in a state of janaabah [major ritual impurity], then purify yourselves...} [Quran 5:6] Scholars of Tafseer (Quran interpretation) said that the verse means, “When you wish to perform prayer...”

Hence, this woman is not obliged to perform ghusl or ablution unless she wants to carry out an act of worship requiring ritual purity, such as prayer or touching the mus-haf (the Quran), for instance.

Allaah knows best.