There's no such a canon of the Armenian Church that prohibits a non-Armenian to be ordained as a priest, but there are indeed people among clergy who say so. However, we have a priest in Armenia, Fr Nshan Panfyorov, who is Russian (father is Russian, mother- Armenian). There's another priest too who, as far as I know, is ethnically Russian. One of the Armenian patriarchs of Jerusalem of the 19th century, Patriarch Harutyun Vehapetyan , was ethnically pure Arab, from Muslim parents. He lived in Cairo until the age of 7 and was a poor orphan. The Armenian Patriarch of Jerusalem, Kirakos, if I remember correctly, once visited Egypt and when being in Cairo, met this Muslim boy who helped the Armenian clergy through minor services. When the patriarch learned he was an orphan, asked him if he would like to go with them to Jerusalem to live with them, the boy agreed. So he went to Jerusalem, learned Armenian, was baptized, given a new name, then grew up, became a monk, and one day, among all the other members of the brotherhood, he, perhaps the only non-Armenian, was elected as the next Patriarch. So, none payed attention on his non-Armenian roots. And he was loved by many.