Just so. You wouldn’t say “I with Natan Sharansky.” You would properly say “Me with Natan Sharansky.” The presence of wife Julie doesn’t change that. Thus “My wife Julie and me with Natan Sharansky.”
I say it’s a gray area, with the correct pronoun’s depending on the implied verb.
It could be, “My wife Julie and I (meet) with Natan Sharansky,” or it could be, “(This is) my wife Julie and me with Natan Sharansky.”
More to the point, it should be, “My wife, Julie, and (I/me) with Natan Sharansky.” Leaving out the commas implies that Julie is not his only wife.