Yes it does matter. God refers explicitly to the Judeo-Christian God and NO other.
Uh, God is the supreme being, so even if the framers of the Constitution may have (may have, because we really aren't sure) intended it to be the Judeo-Christian "God", the "God" referred to is the same almighty supreme being of most mainstream religions. Even though muslims initially were worshipping Allah as a moon-god, the most powerful of many of their pagan gods, I think they generally worship Allah today as the one almighty God.
Were you on the O.J. jury?