is the west an extension of the old roman empire?

In that the Roman Empire adopted Christianity as a way to quell rebellion and maintain in control of the known world - and the Pope chose Ireland as the last bastion of Christianity in 675 - both Christianity and Western civilization did indeed become extensions of the old Roman Empire.

At the time when Ireland became the cradle of western civilization in 675, Rome was in peril of being sacked by the Arabs and Ireland was seen as a safe haven for the continuation of Christianity but also, through it, Roman rule.

By 675, the entire Celtic Pagan population of Ireland had been converted to Christianity by Roman priests. It was more Papist than Rome itself!

In the safe and strongly Christian island of Ireland, western civilization found a fertile cradle but Rome remained in control of the administration of the Church in Ireland - and still does - and at the time, in the 7th century, it effectively controlled the entire country, too.

It was a Roman church system, not an Irish one, and - just like Rome ruled the later years of Hellenic civilization with its cradle in Greece - Rome ruled the early years of Western civilization with its cradle in Ireland.

Without the birth of Islam in 622, the birth of western civilization in Ireland might never have occurred; and, without the adoption of Christianity, Rome might never have remained a seat of power.