I was having a dinner table conversation and a relative in the military was recounting his time serving in Iraq. He described an incident where during a raid a woman started talking to my relative in Arabic and when he looked to the Iraqi interpreter to get a translation of what she said, the interpreter turned and smacked the woman and led her out of the room. The Iraqi man came back and apologized that she spoke to my relative (a man). My question is, considering Islam started about 600 years after Christianity, would a similar thing have happened if a Christian woman spoke to a man in medieval Europe?
As far as I know of, no, although it's possible that I don't know of a source where such a thing occurs. There are certainly sources in which women talk to men freely, though.
One thing to keep in mind is that some aspects of gender roles that might be thought of as Islamic might really be specific to specific Arab cultures (perhaps including Arab Christians).
In Marjane Satrapi's Persepolis, for example, she describes how Kuwaiti refugees in Iran tended to assume that any Iranian woman walking the streets was a prostitute -- an attitude which was rather shocking to Iranian sensibilities, even though Iran is at least as fervently Muslim as Kuwait. As another example, in Tuareg culture, men usually wear veils, and women usually don't.
Usmah ibn Munqidh, in his autobiography, generally gives the impression that the Franks in the age of the Crusades didn't have particularly strong conventions against women talking to unrelated men:
The Franks are void of all zeal and jealousy. One of them may be walking along with his wife. He meets another man who takes the wife by the hand and steps aside to converse with her while the husband is standing on one side waiting for his wife to conclude the conversation. If she lingers too long for him, he leaves her alone with the conversant and goes away.
My impression, though, is that Christians and Muslims in their propaganda about each other in this time, both tended to take the line of "the other side's women are licentious sluts," rather than one side saying "the other side's women are licentious sluts" and the other side saying "the other side is super-prudish and oppressive toward women." (However, I would not consider Munqidh's writings -- whatever their accuracy -- to be propaganda.)
I'm not sure about the norms for women in, say, the Byzantine Empire, or how much norms for women changed between the early middle ages (your question) and the age of the Crusades.