I would assume a high EV would correlate strongly with a high BABIP. Obviously, there's the caveat that the ball is kept in the park, since extreme flyball power hitters could have a bunch of hard hit balls that are homers and thus don't count towards BABIP. I don't think many pop ups are being hit at 100+ mph whereas line drives and hard ground balls can be and the latter two correlate with BABIP. However, I believe there's some interesting results for low EV as well which could be much more luck than skill, i.e. a player that hits a bunch of bloops, high choppers or dribblers that squeak through could potentially have a high BABIP but very poor batted ball measurements. I believe Billy Burns exhibited this in Statcast's first year in 2015, and some early results indicated a sort of valley between the very low EV and very high EV players, not sure how the graph looks nowadays that we have a few years of data.
As Laika points out below, BABIP isn't actually factored in at all in EV and vice-versa, but they're related.