The Church Jesus founded was not discriminatory, but for all; "Go therefore teach all nations..." so a Church that does not both claim to be worldwide and is not actually worldwide cannot be the one Church which Jesus founded.
That is why the Church has been called catholic since the earliest centuries. Even the Apostles' creed makes 'catholic' a part of Christian faith.
The "Roman" bit,comes merely from the fact that both Sts. Peter and St. Paul were martyred in Rome. Paul went there in chains, but Peter went there intentionally to set up his headquarters in the centre of the current empire, so that the message of Christ could be more easily spread throughout the world. The successors of Peter are still bishops of Rome. Hence "Roman" and "Catholic" are the signs of the genuine Church of Jesus Christ.