The answer is No and the reason is the equivalence principle which says that there exist natural units in which the gravitational mass (the mass $m$ in $F=GMm/r^2$) is equal to the inertial mass (the mass $m$ in $F=ma$) for all objects in the Universe. This is equivalent to the statement that all objects, regardless of their composition, density, and other properties, accelerate by the same acceleration in a given gravitational field (any gravitational field).
This equivalence principle is the starting point behind Einstein's general theory of relativity which describes gravity as the curvature of spacetime (and which is fully respected by more modern theories of gravity, especially string theory). This principle is also experimentally verified at the relative accuracy level of order $10^{-17}$ which is incredibly accurate (one may compare two different materials which have noticeably different densities etc. and the acceleration is still perfectly the same). In principle, it's always possible that some experimental deviations will be found by finer experiments in the future (but that's true about any insight in physics). However, it's not just the absence of any experimental hints that undermines any idea that the gravitational force should depend on anything else aside from mass; it is also the complete absence of candidate theories that would be compatible with the basic observations and where the deviation would be implanted as anything more than just an ugly, partially inconsistent, unjustified, and numerically small deformation of a beautiful, consistent, robust, justified theory.
The question whether two liquids mix or not is mainly to do with the intermolecular forces, not the densities. These forces lead to the bulk property known as chemical potential.
Assuming the experiment is done in a pressurized cabin, then the ambient temperature and pressure may take ordinary values such as STP (room temperature, 1 atmosphere of pressure).
At such ordinary temperature and pressure, what happens with oil and water is that the fluid consists of two parts, one of which is mostly (but not completely) water and the other is mostly (but not completely) oil. The entropy of system-plus-environment is maximised when the Gibbs function is minimised, with the result that the oil is not fully dissolved but rather the liquid consists of these two parts. The two parts will gather at separate locations so as to minimise any boundary areas where there is surface tension, and also in response to any ambient forces such as gravity.
In the absence of gravity, therefore, one expects a blob of one liquid and a blob of the other. There are three types of boundary: water-oil, water-air, oil-air, each with a different surface tension. I think the water-air boundary has the highest surface tension, which suggests the oil will surround the water so as to elliminate this boundary. Therefore if there is enough oil then I expect the equilibrium configuration is a sphere of water surrounded by a spherical shell of oil. At smaller amounts of oil I guess the oil will be spread out over the surface of the water, not quite covering it.
Perhaps the configuration with the least Gibbs potential is some other shape (depending on the proportions of oil and water), but the equilibrium configuration will certainly consist of two parts with different concentrations as I have said, unless the temperature is high. At high temperature the two fluids fully dissolve into one another and then you would have a single continuous substance, not two.
Best Answer
It's because the densities of the atoms changes.
The density of an atom is a somewhat vague concept because an atom doesn't have a sharply defined outer surface. nevertheless you can define a size based on the average lengths of bonds formed by atoms, and if you do this you find the size of atoms decreases along a row in the periodic table even though the atoms are getting heavier. This article has a nice diagram showing the size changes.
For example, the density of tetrachloroethylene, at 1.62 g/cm$^3$, is higher than water mainly because the chlorine atoms are denser than oxygen or hydrogen atoms.