Only in an extremely loose sense.

I have mixed feelings about this kind of statements (just like, for instance, the idea that the Ottoman Empire is the logical continuation of the Roman Empire). They can be useful in order to challenge assumptions and to encourage creative thinking; but in the end, they invariably are cleverer than they are accurate. It can be said, however, to borrow Chris Wickham's title, that Christianity (and particularly the Catholic Church) is the inheritance of Rome. Here is just a couple of more substantial answers to your question:

1. The borders of the Catholic Church, even in the immediate post-Roman world, do not match that of the former Empire. In some places, “Roman” Christianity is a good test of Roman-ness: for instance, its nigh-disappearance in south-eastern Britain coincides with a destruction of Roman structures. In other places, the attraction of the Roman model explains the adoption of Nicene Christianity rather than Arianism (e.g. the conversion of the Franks). However, the model is far from water-tight: for instance, the conversion of Ireland seems to have happened entirely after the collapse of the nearest Roman province, and cannot be readily explained in terms of Roman influence/overlordship.

2. Papal use of a quasi-imperial symbolism can be put aside, since it appeared rather lately. Bishops, on the other hand, had been associated with the imperial governement for some time (e.g. Ambrose of Milan), and they proved very important during transitional periods. Interestingly, it can also be shown (at least where our prosopographical documentation is not too poor) that high ranking clergy, for instance in Francia, frequently came from the senatorial aristocracy. However, except during brief periods of anarchy, they never were the only authority in place.

3. The clergy did not think of itself as a continuator of the Roman Empire. It is often said, for instance, that Augustine's City of God articulated (after the first fall of the city of Rome) the model of a Church in a non-imperial world (while the official success of Christianity in the 4th century indeed tended to make the Christian world co-terminous with the Roman world). Conversely, many people thought of themselves as secular successors to Rome. The Eastern Roman Empire, well, was indeed a living part of the Roman world, and it still hosted a church that was originally no different from its Western counterpart. Were there two empires in Constantinople, then? Similarly, in the West, many secular powers immediately took over from the Empire, claiming in many cases to hold their power from the emperor in Constantinople.

I could continue on and on (and would be pleased to do so if you happen to have in mind some particular argument about Roman-Catholic continuity). Overall, I think it is perfectly fair to say that the Empire was a formative frame for the Church, and that the Church did play the role of a preserver for many imperial traditions and motifs. However, the argument cannot be pushed further than this without stretching the evidence enormously.