Prohibitions by the Church of the sort you suggest would have appeared in the canons (or "rules") of the Church councils - either of local councils, convened by bishops in specific regions; or Ecumenical Councils, at which bishops from all five ancient Church Sees (Rome, Constantinople, Alexandria, Antioch, Jerusalem) attended.
The Church was more or less united for the first 1,000 years of its existence, with the exception of the Nestorian and Monophysite schisms in the 5th century.
During the first millennium, the so-called "Chalcedonian" Church on the other side of these schisms held seven Ecumenical councils, from 325 to 787. In addition, numerous local councils were held within the individual sees (e.g. Council of Carthage in 397). The canons of all seven Ecumenical Councils and a number of important local councils have been translated from Greek into English as part of the Nicene and Post-Nicene Church Fathers series.
There are almost 200 canons that came out of the Ecumenical Councils and an even greater number that arose from the various local councils. There are no canons from any of these councils that expressly forbid marriages between a man and a woman of different races or nationalities.