Yes, it is fair to characterize the Gothic subculture as being under the umbrella of the post punk culture. In some ways both cultures are very similar and take on the same visual effects. Some of the cultures of both groups are the dark clothes and piercings. It is not that hard to put these two cultures under the same umbrella.
The punk culture characterized the underbelly of the 1980s. If that's true, then the Goth movement typified the same subculture of the 1990s. Just like the funk of the 1970s was a reaction to the free love of the 1960s, one subculture influenced the next. Punks and Goth subculture are essentially the same as they are minority subcultures among teens and young adults. Goth subculture is definitely in the post-punk culture as the 1990s were after the 1980s.
It is fair to characterize the Gothic subculture as being under the umbrella of post-punk culture. The Gothic subculture began in England in the late 1970's and early 1980's. In 1981 an article published by a UK magazine called "Sounds", was titled "The face of punk Gothique". In this article the author asked whether the new Gothic subculture was arising, and born out of the post-punk culture which was the predominant musical culture in that time.
Young people today have many subcultures, and even the practitioners themselves often do not know how to define themselves. It does not profit anyone to worry about semantics and the nomenclature of these different groups. Anti-social groups, by their nature, do not want to be discussed by the mainstream. We should leave these groups alone.
No, it is not fair to characterize the Gothic subculture as being under the umbrella of post-punk culture, because they developed in different ways. People who are Goth are different than people who are post-punk. Most notably, they even dress differently. They try to make different statements. They are both counter culture, but beyond that, they are very different things.