A frustrating read with the use of gender rather than sex. And, tbh, given that gender reassignment is a protected characteristic, I rather think he didn't have a leg to stand on here. I'm sure he doesn't agree with homosexuality, or gay marriage, but he would still have to acknowledge the partner of a gay patient.
I'm sure many doctors disapprove of the choices their patients make, but conscience clauses are for things like life or death stuff, abortion, euthanasia etc. Not just stuff you feel uncomfortable about.
However, if he wishes to make a principled decision that he cannot do the job because it goes against his conscience, then he is entirely within his rights to do that and I'll not criticize on that score.
Oh, and as an aside, I couldn't give a monkey's that it's the DM. I get the simple snobbery about it, but I despise the various campaigns to shut it down because of an editorial stance unpopular with some people. If there are views you don't like in the public square, you refute them, you don't say "Ew, those political views are too, too horrible. Why can't everyone read the Guardian!" Because censorship is fine and dandy when it shuts down the stuff you don't like, but it's an uncontrollable genie, and once you rub the lamp, it won't be 'reasonable'.