We can't really speak of an empire by the time of the death of Muhammad, but the Constitution of Medina and the Qur'an, the two most important sources on Muhammad, speaks of the Ummah (community, nation) or Ummah Wahidah (one community/nation). The usage of the term is not totally consistent and evolves over time, as one would expect from such a generic term, but in the CoM, it appears to be religiously inclusive of Jewish tribes and so probably the closest to what you are looking for. In later sources it is typically synonymous with Ummat al-Islam, "the Islamic community". Since in Muhammad's time we're speaking of a network of tribal allegiances rather than an administrative empire, not that much terminology is necessary.

Muhammad's perspective on and intentions behind his conquest are matters of widespread debate; two good popular works I've read and enjoyed are Muhammad and the Believers by Donner and In God's Path by Hoyland.