Is the reason for prohibiting greeting a woman out of fear of temptation, and is it forbidden in and of itself, or for something else (adultery), especially in a European country where not greeting is considered rude? What is the Sharia ruling on this matter?

Greeting a woman verbally is permissible when there is no fear of temptation, but it is disliked if there is such a fear. As for a man shaking hands with a young, non-mahram (definition) woman, it is forbidden according to the majority of jurists. Some of them qualified the prohibition by stipulating that she must be desirable. It is obligatory to avoid shaking hands with non-mahram women and to adhere to the limits of the Sharia (definition), prioritizing the pleasure of Allah over pleasing people.