History of Feminism

In consent culture, he absolutely is allowed not to shake hands with anyone he doesn't want to shake hands with. Many cultures don't touch when they greet each other. As long as he does something (a nod, a bow, a smile) to acknowledge the person he is greeting, I say ok. He cannot refuse to be served by a woman either, when the services don't involve nudity (doctors) or food.

As for discriminating, that's another matter. Some positions are inherently discriminating - Catholic priests, for example. Others should not be. Jobs driving buses around ultra-Orthodox Jewish neighborhoods, for example. They get licences from the city, so no gender discrimination there. In public transport, like a plane, if you want to make sure you don't sit next to someone specific, buy the tickets of the relevant seats. Don't ask women to move to accomodate you, pay up, or don't travel.