One can generally ignore the effects of gravity when dealing with anything human scale or lower, but for our purposes, let's assume 2 1 kilogram plates space 10 nanometers apart, F=-GMm/r²

With r= 10^-8, r²= 10^-16, with M & m = 1, F=Gx10^16, G=6.67 x10^-11
Net Force there is thus F=6.67x10^5 Newtons

LEt's say those plates are 1 kg sheets 1 meter square each. This Force would amount to 6.67x10^5 Newtons per square meter, or 66,700 N/m², or 66.7 kPa, kilo-pascals, normal atmosphere is about 100 kPa, the normal Casimir effect at 10 nm incidentally.

Now, those two seem kinda close, but that's because I used 1 kg plates. Had they been 100 gram plates it wouldn't be 66.7 kPA, but rather 667 Pa, 100 times weaker, yet you'd still have that 1 atmosphere or 100 kPa force, Use thin foil of a few grams and you'd get it too, or alternatively use monstrous thick plates like two bank vault doors. In all cases you'll get that same 1 atm pressure at 10 nm.

Second, IIRC Casimir falls off with the Fourth power of distance, whereas gravity is inverse square. Double the distance between two objects and the gravity between them drops by a quarter, double the distance for Casimir force and it drops by 2^4, or 16 times less force.

So you see, gravity is easily reduced from the equation. Double the mass of both plates and force increases from gravity by a factor of four, but Casimir force would remain the same, double distance and gravity is quartered whereas casimir is reduced to a sixteenth.

Conceptually easy, experimentally a pain in the ass, strictly from real world practicality issues, but it's been done.