An "inquisitorial" system is one where the Judge or Magistrate actively questions the accused and witnesses to attempt to determine the facts. The Judge may also determine, at least in part, what witnesses to call in what order. An "adversarial" system is one in which each side presents its case, and the judge acts as an umpire deciding on procedure, and possibly makes the final ruling (or directs a jury to do so) but is not actively involved in questioning witnesses or deciding what witnesses to call.
I don't see anything which would prevent a common-law jurisdiction from establishing an "inquisitorial" system by statute except longstanding tradition, but as far as i know no such jurisdiction has ever had such a system in place for dealing with criminal matters. The informal procedures in some small claims courts do have judges more actively involved than in other courts. I think this is also true in some family courts as well.
I think I have heard of some civil-law jurisdictions which use something like an adversary system, but i am not sure of that. Certainly a civil-law country could pass a law setting up such a system if it chose to.