Unlike Spanish, English has a strong, widespread tendency to use abbreviations whenever possible. No such thing happens in Spanish. We could say, practically as a joke to imitate the English abbreviation, things like: "Vive en USA," but this is only allowed in colloquial language. Also, unlike other abbreviations like UK, the initials "USA" form a word that can be pronounced in Spanish as if it were a word in the language.
Surfing the Internet, I just confirmed my suspicion that "Dominicana" can be used as a short form for "República Dominicana."
Notice, however, that the official name is as specified here. In documents from that country I have often seen "R.D.", but this will occur in writing, not in speech.