The current ABA does seem to stake a claim to the history of the old ABA. The new ABA is a league that has over 50 franchises operating in all sorts of locations throughout the USA and around the world. Some of the cities also share an NBA team, but most are from markets not served by the NBA. The league has a wide variety of logo styles for its teams, some really good, others look like the team owner had their little brother design them. Many of the logos contain an image of the ABA red, white and blue basketball, in the great tradition of professional basketball logo design.
So, now you know that technically the ABA still exists. The league will never see the likes of a Dr. J of George Gervin gracing their courts, but a small piece of the spirit of the old rebel league still exists. You can find more information about the modern ABA here at their official website.