I guess you must be a woman then?
If so, I guess he must be Muslim, in which case he is correct. Physical contact between any unmarried man and woman in Islam (& also Orthodox Judaism & possibly Buddhism) is forbidden. This includes handshaking.
Like any religion, different people take it on to different degrees of strictness. Many modern Muslims living in the West understand that hand-shaking is a common greeting especially in business and thus relax this rule, but some people really cannot do it. This article is interesting:
http://www.rukhsanakhan.com/article-etiquette. htm
I think you need to respect his wishes, plus although it sounds weird, if you look at it from his POV he is being respectful towards you by NOT shaking your hand.
If you turn him down for the vacancy because of this factor alone (without taking into ******* his experience, qualifications and ability to do the job as advertised) then you could possibly be accused of religous discrimination.
If he's not suitable for other reasons too, then don't worry about it, but he may be perfect for your company, you never know!