Can you clarify?

What do you mean with 'Rome's version of Christianity'? The one today known as Roman Catholicism? The Christianity dominant in the West? The one dominant in the East? And in which period (especially for the East).

The problem is that it almost impossible to define 'Rome's version' of Christianity, or even the 'Roman version' of Christianity since a single version or 'Roman' Christianity did not exist at all... In the period in which Armenia became Christian (or at least adopted it as a state religion), there were many tenets of Christianity, and amongst other battles there was for instance the battle between Arians and 'orthodox' Christians - the former probably being the majority in the East, but the latter favoured by Constantine (a major reason for them being the winners at Nicaea), and later on by Theodosius. Indeed, there were many 'versions' of Christianity almost from its inception.

It can be questioned to what extent even the Roman Catholic Church (as it has become known) as such was homogenous at any point (from Antiquity through the Middle Ages into today).

Odds are that at the moment Armenia adopted Christianity - due to the conversion of its king in 301 - as its state religion it adopted the doctrine of the missionary converting the king (similarly to how the Goths came to adopt Arian Christianity). To what extent this version actually matched with the version of Christianity that became official in 313/325 let alone under Theodosius is probably impossible to determine (due to a lack of information), though it was probably not too different from the 'orthodoxy' established at Nicaea in 325 (as apparently the Armenian Church agreed with its creed).

Would Christianity have become as big in Ethiopia as it did without the Romans (or I presume the Roman Empire)? Probably not, though - of course - we can never know.