Generally, if someone says that they are an orthodox Catholic, they mean that they fully accept all of the teachings of the Church. Ordinarily, the “orthodox” part of this would not be capitalized unless it were at the beginning of a sentence or part of a title or something.
A person who is describing themselves as an orthodox Catholic usually really means that they are an orthodox member of the Latin Catholic Church (because that church is much larger than all of the other Churches put together). However, it could just as well mean that they are an orthodox Melkite Greek Catholic or an orthodox Maronite Catholic or whatever.
The term “Roman Catholic” is a little tricky anyway, because some members of Eastern Catholic Churches are offended to be called by it, and some are proud to be called by it. It depends on what you mean. All of the Catholic Churches are in communion with the Bishop of Rome (the Pope). But they are not part of the Latin Catholic Church, which is also based in Rome.
Anyway, in answer to your original question, no. In this context, “Orthodox” is generally just a descriptive word, that is similar to saying “Faithful Catholic.” In days when so many of our politicians label themselves as Catholic without following the teachings of the Church in their public life, and many people are Catholic but don’t even
know many of the teachings of the Church, people sometimes want to have some way to say “I’m a Catholic by belief, conviction, and actions, not just by Baptism and/or national heritage.”
The best thing would be if all Catholics followed the teachings of the Church, and then you could say Catholic and it would
mean orthodox Catholic.
HOWEVER, it is also true that the Eastern/Oriental Orthodox Churches which are not in communion with Rome officially refer to themselves as Orthodox Catholic Churches. Generally there would probably be some other descriptive word in there to indicate which of the Orthodox churches were meant, e.g. Russian Orthodox Catholic or something. Anyway, if the people are calling themselves Orthodox Catholics in that context, then certainly Orthodox Catholic wouldn’t replace Roman Catholic, without a whole bunch of people (say a billion or so) converting.
–Jen