What is the ruling on a male doctor seeing the 'awrah if there are female doctors available? Is this considered one of the major sins? Is expiation required for that, and how does one repent, especially since this occurred in Ramadan?

If a female specialist physician is available for the woman's condition, even if she is not Muslim, it is forbidden for the woman to go to a male doctor. The prohibition increases if the doctor is a disbeliever. Sheikh Ibn Uthaymeen clarified that it is not permissible for a woman to go to a male doctor if a female doctor can treat her, especially in matters of 'awrah (intimate parts), because uncovering the 'awrah is not permissible except out of necessity. Similarly, a male doctor should not accept women for treatment if there is a female doctor who can perform the task. As for the confusion that led to the uncovering, which you both experienced, it does not invalidate the fast. You both should seek forgiveness and repent and be careful in the future. There is no expiation required of you; rather, it is prescribed to follow a bad deed with a good one.