Oof. It's a matter of symantics. The Western Wall is not a part of the inner Temple where the altar and the ark of the covenant were. It's part of the outermost wall that held up the Temple. I'm going to quote a block from the link I already posted (http://english.thekotel.org/content.asp?id=212)
The Western Wall is part of a big renovation project initiated by King Herod.
In the year 37 BCE, Herod was appointed king in Jerusalem and he soon initiated a huge renovation project for the Temple. He hired many workers who toiled to make the Temple more magnificent and to widen the area of the Temple Mount by flattening the mountain peak and building four support walls around it.
The Western Wall is the western support wall built during this widening of the Temple Mount Plaza.
What makes the Western Wall (and not one of the other three remaining support walls) the most special is its proximity to the location of the Holy of Holies in the Temple.
The Second Temple was destroyed in the year 70 CE. Despite the destruction that took place, all four Temple Mount support walls remained standing. Throughout the generations since the Temple’s destruction, the Western Wall was the remnant closest to the site of the Temple’s Holy of Holies that was accessible to Jews. Therefore, it became a place of prayer and yearning for Jews around the world. When Jews expressed their longing for Jerusalem through song, Judaica, jewelry, and prayer, the image of Jerusalem was conveyed via the image of the Western Wall.
You could claim that the wall isn't part of the Temple, but it's part of the foundation of the Temple. It's up to you to decide whether a structure that supported the Temple is itself part of the Temple.
Here is a link to what the Temple looked like. The arrow points to where the Western Wall is. "The Temple" itself is all above that wall, but the Wall is part of what holds up the Temple. If you click through and look for the image on the page itself, it's next to a picture of the same position in the modern time.
You could make a case, and it seems your friend is doing so, that it's not actually part of the Temple ground. My opinion is that that wall is part of the Temple.
