You’re not under any obligation to wear that badge if you don’t want to, and as far as I’m aware you have to go on a specific training course to get one anyway as wearing it means you have been trained on specific issues to do with lgbt healthcare.

What I would say though, is that being lgbt is not a “movement” - it’s a fact of people’s lives that is inescapable and not something people choose. It is simply ignorant to insist otherwise. Sure there are movements to further the rights and wellbeing of lgbt people such as the rainbow badge initiative but no one’s going to force you to take part. However lgbt people exist as colleagues and patients whether you like it or not, and deserve basic respect. I get the sense from the language that you used to describe your colleague that you find his openness about being gay distasteful, and I wonder if he senses that too, hence being distant.

To be honest it just sounds like your values are homophobic. You have the right to your own opinions and thoughts but at least call a spade a spade.

A disclaimer: I’m a doctor and also a gay person, so of course I am biased here. I do want to say that people in my position literally cannot choose whether we are lgbt or not, it is just something that is part of us and that we live with. I don’t tell anyone at work that I’m gay specifically because of reactions like yours to your colleague, no one wants to jeopardise their working relationships.