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Like Dr. adequate cited, the force of gravity between two objects is inversely proportional to the distance between them.
You probably know this, but it is important to remember that it is the inverse square of the distance.
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Like I mentioned in my original message, this action from all masses from all sides has the affect of holding and securing the original object in one place-at the center of the universe
No, no, no. Firstly this really only applies if the mass is evenly distributed. Since this is not the case local concentrations of mass will have a far greater effect (thanks to the inverse square law). Adding Relativity makes it worse because even Special Relativity denies that there is a meaningful "centre" to the universe.
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This universal force provides inertia to the object which has a value we call mass.
No. Inertia is another thing entirely. It is not a force at all as your idea would require. Again you only need Newtonian mechanics to understand this.
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...why isn’t the mass of the object changing in response to the change in the distances of an expanding universe
It hasn't been checked because it makes no sense. Weight - not mass - is the effect of gravity. And that is usually dominated by local mass concentrations. On the Earth's surface weight is overwhelmingly due to the attraction of the mass of the Earth - the rest of the universe has a negligible effect in comparison. (True, the moon has enough of an effect to cause tides, but even that is relatively small - and other bodies, even in aggregate, have even less effect).