What is the ruling on a doctor shaking hands with women in the workplace if refraining from doing so leads to threats of administrative procedures being halted or the contract being terminated?

It is absolutely impermissible to shake hands with a non-mahram (definition) woman, even with a barrier, due to the temptation and arousal of desire that may result from it. This is supported by the prophetic Hadith (definition): "For one of you to be stabbed in the head with an iron needle is better for him than to touch a woman who is not lawful for him." Moreover, mixing in the workplace is forbidden due to the proscribed acts it entails, such as shaking hands, looking, and being in seclusion. It is obligatory for a Muslim to flee from these forbidden acts, even if it leads to leaving the job. Lowering the gaze from non-mahram women is obligatory, except for a sudden glance.