Apart from its meaning as a political unit, the English word "country" also carries strong denotations of "land, terrain, territory", and somehow a country seems earthbound to me and nothing that might span several planets or even solar systems.
Of course there are words like "state" or "nation", but these don't have the same connotations of "home" and "belonging" and are not used in the emotional sense that the word "country" is used.
For example, assuming Patrick Henry would have lived in the United States of the Orion Arm instead of the USA, what word would he have used in his famous quote instead of country?
Show me that age and {country} where the rights and liberties of the people were placed on the sole chance of their rulers being good men, without a consequent loss of liberty?
Or with what word would George S. Patton have called his fellow citizens to arms?
... the highest obligation and privilege of citizenship is that of bearing arms for one’s {country}.
What would "interstellar countries" be called? Would the meaning of the word "country" expand to include this new kind of national identity? Would new words have to be made up? Or does English already have a word for this?
Note: I'm looking for a term that is independent of the type of government and might apply to a democratic state, so "empire" or "realm" are not the best fit.
Edit:
"Country", to me, is the term that people use in an emotional context. When they cheer for their national team in the soccer world cup, then "country" is the word that they think of, not Kingdom, or Empire, or Federation, or State. To an American, the USA is their home country, not their home federation.
In the British Empire, things were a bit more complicated. I think that England and India were considered countries – they were the home that people identified with. People from the British Emprie might have had some kind of double identity: as British and as Indian, for example. Within the Emprire they might have been proud to be Indian, instead of Australian, but at the same time they might have been proud to be British, not Chinese. Interestingly enough, the UK doesn't have a national soccer team, but England, Scotland and Wales do. So Wales is a country, while the UK is not (I think).
Looking at the British Empire and the UK as an example, it seems to me that my question is contradictory. Very likely, as KeithS wrote in his answer, "a planet [would be] considered similarly to a country on Earth", and a political unit spanning several worlds would be something like an empire or federation on Earth. And people from such an interstellar empire or federation, would consider a world (that is, a planet) their home, not the empire, unless they were faced with a person from outside this empire.
So I think, the best answer to what would correspond to a country in outer space is in fact a world, or planet, or, if planets are divided into smaller political units, a country, just like on Earth. A unit spanning more worlds would probably not instill the same feeling of belonging as a single planet would, and would not be considered a person's home and not be called a country.
Thank you all for your valuable feedback. This was very helpful. (I'm writing a novel and now I can write on.) I upvoted all answers, because I found them all helpful. I'm choosing the answer by inappropriateCode, because it best reflects my own opinion on the matter after reading all the answers and thinking on them. I find the answer by Psychrom most intriguing, but it does not completely apply to the problem at hand (emotional attachment to one's origin).