There's really not a response that fits what I think I'd do in reality. I would not participate in the circle because I would feel it was hypocritical of me. I hope I would have anticipated the pastor's suggestion and bowed out. I don't think that is the time to announce my religious status. I am not even sure they would notice. Their minds are elsewhere.
It is a problem, however. We should not have to hide our disbelief, but there are some real-world implications that we would have to be aware of before we 'come out', particularly if we're involved in a vocation that relies on customers (or patients) to choose us.

Say "we will do our best. I must get to surgery now."

Been reading through some of the responses. I'm kind of surprised so many people are willing to "go along" with the delusions of others.
No wonder we can't get the influences of religion out of our lives.
Too few people are willing to tell these people "No, thank you".
The setting and situation doesn't matter.

Participate in the ritual. It certainly won't harm you. Giving comfort to patient, their kin, and as a team building ritual..... do it.

I'm really tired of believers thinking everyone is as delusional as they are, and expecting complete strangers to participate in their delusional rituals.
I do not respect their beliefs and I'm not going to be forced to pretend that I do, regardless of the circumstances.
Enough already.

I would hold hands, but l wouldn't bow my head.

Since most surgeons think they're god, seems like you'd think they were preying to you.

I would state, "Sir, am not religious, but I will do everything in my power to ensure that Mr. (or Mrs.) X has the best of outcomes." Then I would step out.

Wear a rainbow necklace that says "Pride" to avoid even being asked.

You can smile and excuse yourself during this time, or just play along and think of it as positive thoughts. You can also use the time to meditate, which is always good.
No need to shut your eyes, though.

“I respectfully decline.”

Sometimes ya just gotta humor the poor lil things like the children that they are !

I would explain I'm an Atheist but would be willing to join hands with them in my own thoughts if they welcomed that.

I think the time has finally come that we all step up and step out of the closet. Otherwise, we will be over ridden by the religious extremists.

I do work in healthcare and I have had this situation,
You show respect and compassion for the family and join in.
This isn’t about our beliefs. It’s about a group of people that are scared. We are the professionals and it’s our job to guide explain and help.

This scenario is nothing compared to what we deal with on a daily basis. Shit a large amount of the things I’ve read on the different boards here defy logic and common sense.

We have have different ways we interpret the environment around us: odd beliefs or rituals, religious ideals, social habits. None of us are guilt free of such things. Why be so critical of someone wanting to pray?

Hell yes I’ll join in, because it’s considerate to the family. They are scared shitless and rightfully so. It’s our job in healthcare and if that’s confusing or a problem then do something else for a living.

I'll hold hands, nothing else.

Let Jesus guild the knife of the Doctor and then let him steal your soul.

Not!!!

You say, ok ill go get your Mom. You have your god and prayers so there's no need of us. Have a nice day.

Explain i am there to do a job and that is and shouldnt get distracted from it politely whilst in my head think get the feck put of my way and let me concentrate on the important shit

Tell him/her I am not a believer, but I will give my patient the full benefit of my training and discipline, then I would excuse myself and leave the room.

Say "No thank you, I'm not religious" and stand aside respectfully until they're done. No need to leave the room, and certainly no need to engage in other demonstrations of dissent.