(although I've never understood why they feel a need to apologize for not knowing halacha...).
Baby Einstein, this is what I meant. Most religious Jews, especially those living outside of major Jewish centers, do not expect others to know the details of what we do and don't do and therefore do not take offense to something like shaking hands. If you were to offer to shake, he would probably just say something like "I don't shake" and move on. Most people also are not bothered by your asking why, as long as you are really curious. I've had my fair share of people come bck with nasty replies and trying to convince me that I'm backward and unenlightened for not wanting to shake--those responses are, to me, unenlightened.
The basic thing to remember when dealing with religious Jews is that we believe that our laws are given directly from the mouth of G-d, and will not bend them to suit our needs, no matter how inconvenient. On the other hand, not every issue is clearly written out in the Torah, and the Rabbi's are the ones who have the tools to understand how to apply the basic laws (there are 613!) to each situation. For example, it took years to determine whether we could use electricity on the Sabbath, as electricity is clearly not mentioned in the Torah. The Rabbis who worked on that went to the leading scientists of the day, learned physics, obtained a real understanding of what electricity is and how it works and used the tools from the Torah to determine that we cannot use it on Shabbos.
The same is true of medical situations. There are many Rabbis today who "specialize" in medical halacha (Jewish law). Some of these Rabbis could almost be doctors themselves! For example, my father's Rebbe, known as the "Bostoner Rebbe" bec. he lives in Boston, runs a large organization which assists people with medical referrals, hospitality when they come to Boston for procedures, and funding for those who can't afford to pay. After more than 60 years of doing this, he has a deep understanding of medicine and can discuss cases with leading physicians on nearly an equal status which allows him to determine where a new procedure fits in to Jewish Law.
My suggestion would be that when you run into a conflict with a patient where they don't seem to want to do what you think clearly needs to be done, ask them to get in touch with one of these Rabbis who can help them and you figure out what the law actually is and how it is applied in your case. To an Orthodox Jew, a Rabbi's ruling is as just as binding as G-d's written laws and he will then follow what the Rabbi says. On the other hand, if the Rabbi rules that he should do something different, you may have to be somewhat flexible bec. he will not go against the Rabbi.