The baby was declared dead soon after the operation, while the woman died shortly after.
Police say a breathalyser test has confirmed that the doctor was drunk.
They are investigating whether the deaths were caused by negligence or due to other medical reasons.
Dr PJ Lakhani is a senior doctor with substantial experience and has been working at the government-run Sonavala Hospital for the past 15 years.
The patient, Kamini Chachi, was brought to the hospital on Monday evening with labour pains.
Local media said that the waiting family was told that the baby had died, while the mother was bleeding profusely.
The family decided to shift her to a private hospital but she died on the way, they added.
Police told BBC Gujarati that Dr Lakhani called them, asking for help as he was afraid that the family would attack him once they learned of the deaths. “We found that he was drunk when we arrived so we arrested him,” officer HR Goswami said.
The hospital has formed a committee which will further investigate the cause of her death.