Upon further reflection, I think someone who is arrogant might be more insulted by being called ignorant because they are arrogant. Whereas someone who is ignorant, but not arrogant, might be more likely to concede to a charge of being ignorant.
All in all, I think someone’s reaction to being called arrogant or ignorant depends on their attitude. Since arrogance is more attitude-related than ignorance is, arrogance itself may have more to do with how one reacts to being called arrogant or ignorant than being ignorant.
Anyhoo…
Yeah. What others have said here.
Would you really be offended if, for instance, someone told you that you were “ignorant” on the mating habits of the dung beetle (if you, indeed, are)? Or what if you are told that you are “ignorant” about film music written by British composers prior to 1950? Would that cheese you off (if you weren’t an expert of British film music, that is)? I doubt it.
I plead ignorance on many subjects. I am not ashamed that I am completely ignorant of Pokemon, for instance. No shame there. I’m not ashamed that I am not an expert on opera (though I do enjoy it). I’m not ashamed of that either.
“Ignorant” can be used in many, many non-insulting ways. At least it wouldn’t be insulting to the non-arrogant. 
I’ll admit to being ignorant, especially if my ignorance occurs in areas where I have little experience or interest. I imagine most of us are like this. The only time I feel insulted when someone calls me ignorant is if they are really trying to say that I am stupid. If someone says I am ignorant about professional sports, for example then I accept this, since I really don’t care about this topic. If somone says I am ignorant about sports but does so in such a way as to imply that I am stupid overall, then this is another story. Not knowing who won the Superbowl or even who played in it does not make me stupid.
Being called arrogant, on the other hand, would be an outright insult. To me, arrogance=conceited asshole who is so full of himself. I am quite modest, so I would not only be insulted, but surprised that anyone would ever call me arrogant.
Yes, the intent behind someone calling someone ignorant is a major determinant as to how someone reacts to being called so.
But, how about this situation…
Joe is pretty ignorant about native culture. He’ll freely admit this, and may even wish that he knew more.
During a dinner with friends and friends of friends, he uses the term “Native American”. One friend of a friend - Sue - takes offense to this and calls him ignorant in a somewhat harsh manner.
Joe is embarrassed and somewhat insulted. Because of his ignorance, he hadn’t known that many natives do not like that term. However, Sue was being ignorant herself for assuming that he should know better.
Hopefully, Joe regains his bearings enough to explain that his is indeed pretty ignorant when it comes to native culture, and Sue is decent enough to give him a chance to explain so. Joe could then ask Sue to remedy some of his ignorance, and explain to him why the term he used offended her so.
Then Ted, a real dim bulb, asks Sue if her named is spelled “Sioux”? :smack:
Anyhoo…
I’ve rarely, if ever been accused of arrogance; in fact, I’m rather too often called overly deferential and self-effacing. If someone ever called me arrogant I’d feel pretty proud of myself.
People rarely call me ignorant, either - ignorant in general, not on some specific subject - but if they did, I’d take it as a major insult.