My answer would be no, because the Catholic faith is one, and there is nothing Melkites "don't have to accept." Though, terminology-wise, could be a completely different question. So, as I understand it, Maronites will refer to Purgatory, Immaculate Conception, etc (real Maronites correct me if I'm wrong), while Melkites won't. That doesn't have to be a rejection of the truths they pertain to, though.
There are other Byzantine-rite Catholic Churches besides the Melkites, ya know.
Here are the directories for the three Ruthenian eparchies (dioceses) and the archeparchy in the US. Keep in mind that the eparchy with jurisdiction over an area isn't always the logical one (Pittsburgh has jurisdiction over Texas, for example) and that some eparchies share states between them (Passaic and Pittsburgh share PA, for instance):
Archeparchy of Pittsburgh, PA: http://archpitt.org/directories/parishes
Eparchy of Parma, OH: http://www.parma.org/parishes.html
Eparchy of Phoenix, AZ: http://eparchyofphoenix.org/directory.html
Eparchy of Passaic, NJ: http://www.eparchyofpassaic.com/parish.htm
Also, if you live anywhere near Cleveland, go to the Byzantine Catholic Cultural Center.
It is among Byzantine Catholics. Our Archbishop (that is, His Excellency WILLIAM (Skurla), Archbishop of Pittsburgh and Metropolitan of the United States) is Excellency and the other bishops are Graces.
Well, could be worse for them: they could be Eastern. We just have to hit the hand, which is easier than the ring.
I do (the hand, that is, not the ring #ImaByzCat), but only when a blessing is given or something. One priest I know kind of finds it awkward, I think. Another is probably used to it by now, whatever his personal thoughts on it. Especially given that his mom does it. To him.