The point is that the Casimir effect is an effect that comes out of QED - which you can calculate and everything, and in QED, people didn't put any gravity.
It's a more sophisticated version of a question like: "does magnetic induction have a relationship with gravity" ?
The simple answer is no, because magnetic induction is described by Maxwell's equations, in agreement with observation. Engineers wanting to apply magnetic induction (in, say, a transformer) use Maxwell's equations, and don't use anything related to gravity, and the stuff they make, works.
Of course, then you can come up with "yes, but there might be a relationship between the Maxwell equations and Gravity, Klein recently published something about that".
True. But it doesn't change the fact that magnetic induction is well described by a theory that hasn't gravity in it.
Same with the Casimir effect. It comes out of QED, and QED doesn't contain gravity in any way.