Before assuming the other person "refuses to take constructive criticism" I think the "critic" has to ask himself about himself whether he's the person supervisor at work and/or whether he's the person's teacher/instructor. He should also ask if it's one matter/project/issue on which he has some criticism or whether he's generally someone who thinks he has some right to criticize others on whatever he's inspired to criticize.
Most of us grow up hearing in school that we should (of course) be able to accept the criticism of teachers (and most often, whether that criticism is correct or not, the fact is it is intended to be constructive). A lot of people (most, maybe) don't realize that once we're not in school, or if we're not in a work setting where a superior's constructive criticism (or even another co-worker's suggestion about, say, how a task might be better accomplished), there is actually very isolated and limited use or place for criticism - or at least for the expectation that one's criticism will be appreciated, wanted, or otherwise not found objectionable.
If the "critic" is a customer or client of a business it's reasonable that he voice his complaints about goods/services, and it's reasonable that he offer a suggestion. In personal or other "non-structured" settings, unless a couple of people have agreed that one will teach the other something (and constructive criticism would be reasonable in that setting), criticizing others generally suggests that one person believes he knows better than the other about how to do something or what the other should be doing. A lot of people are more than willing to accept criticism when they've asked for another's ideas on something. Many aren't interested in unsolicited by people who may, in fact, not "know better" or know enough the other person's reasons for doing what he does.
People guilty of inappropriate, over-bearing, "constructive criticism" need to ask if there's the chance the other person is fine accepting appropriate criticism, but just isn't interested in hearing that one, over-bearing, person's "two cents" on some matter.
"Overbearing" can be sweetened up and/or well intended, but most people see through it. Ego is often about who thinks he's somehow superior to others, or who needs to be. Sometimes when two ways of thinking/doing things are involved it's the critic who can't accept that the other person may "know better" about a matter (or himself, his own life).