What is the ruling on shaking hands with a non-Muslim doctor if it is considered a sign of respect and refraining from it might cause a problem, especially when dealing with multiple doctors?

It has been previously clarified that it is forbidden for a woman to shake hands with a non-mahram man. Being in Western countries does not justify this forbidden handshake. A true Muslim takes pride in his Islam and is not ashamed of the Sunnah of his Prophet, peace and blessings be upon him. What the questioner mentioned is not considered a necessity that permits a forbidden act. A Muslim woman's adherence to the rulings of her religion may invite non-Muslims to consider Islam.