What is the ruling on shaking hands with a woman in a foreign country with the intention of not giving her a wrong impression about Islam and avoiding embarrassing her?

Shaking hands with a non-mahram woman is forbidden, and this action is blameworthy and impermissible. The Sharia came to prohibit compromise in matters of religion. Allah Almighty says: (They wish that you would compromise [in religion], so they would compromise). And whoever seeks the pleasure of people through the displeasure of Allah, Allah entrusts him to the people. Steadfastness upon principles is the best way to earn people's respect for a Muslim's religion.