My first answer was philosophical, this one is more Halachic. After spending a couple of hours in our Kolel researching the subject we've come up with some findings:
1. Learning straight from the Torah:
I couldn't find a clear view of Poskim on the subject (if Rambam does not deal with that - nobody does), but there's a consensus of the Torah commentators that Jews are allowed to [judge and] execute NJs as we learn it from Parashat Dina (Gen 34). They agree that all male residents of Shechem were killed by being judged and sentenced to death, they only slightly disagree on what ground.
The problem is that nobody mentions where exactly they were sentenced - what BD if anything. We also see that Yaakov disagreed but unclear on what (but not on the fact the residents were judged for sure).
Another example is Moses killing an Egyptian that beat a Jew. Moses witnessed the action and sentenced him to death and executed him. Here also, nobody argues on the fact that Moses judged and sentenced and executed only the execution was verbal and that opens additional possibilities of interpretation.
The problem with those examples that they were before Matan Torah and we always can claim the laws were different.
3. Learning from Gemmorah
We can learn "circumstantially" from the [very existence of the] Sugya of "אין עד נעשה דיין" B"B 113, that discusses if a Jew who saw a NJ committing a crime can be the [one] judge to judge [and therefore] sentence him.
3. Learning from Rambam
Rambam rules numerous capital offenses for NJ incl. 7 Noahide laws, beating Jews or having relations with Jewish women or for keeping Shabbos, using different languages, such as "חייב מיתה" or "נהרגעליו" etc. But he never mentions how the judgment should be carried out and by whom.
Also, when Rambam rules about our responsibility for establishing NJ courts (see @Joel's answer) in Hilchot Melachim is unclear when it applies, seemingly, only in times when we are able to (or maybe in this time, like the Chabad's initiative of 7 Noahide Mitzvos).
3. Learning from Chinuch and Minchas Chinuch
Chinuch Mitzvah 409 ("one cannot be executed unless he stands a trial") claims interestingly that "those Mitzvos do not apply to NJ". Once again, it is not a clear answer to your question, but from that statement, we can learn קל וחומר as many Poskim ruled (resolving "אין עד נעשה דיין" negatively - one can be a witness and a judge at the same time) that a Jew is allowed to execute an NJ, representing "a whole court" and an executor.
So if one Jew can be a court, an NJ can surely be brought to a "real" court.
I worked a lot on this answer, please respect. Also, please don't downvote it just because you disagree or you don't like the conclusion - use comments.