Is it correct to assume that if something can create a paradox, this something itself cannot exist nor be created? That sounds logical, because a paradox leads to a senseless conclusion, it's unquestionably impossible so it cannot exist. The anthropomorphic omnipotent God can create a paradox, for example "Can God make something he cannot lift?". His omnipotence should allow him to break scientific laws, so I would assume God can't coexist with the totality of the universe.
You're conflating things a bit, so let me try to parse what you're trying to say.
Statements that lead to contradictions can't possibly be true
The existence of God leads to contradictions, so it can't possibly be true
Essentially, you're arguing that because God can create "square circles" -- a logical impossibility -- he cannot exist. Conversely, if he can't create "square circles," God really isn't omnipotent so, again, God can't exist. This argument doesn't work due to a very old distinction between (what Avicenna) called particulars, universals, imaginaries, and impossibles.
A "square circle" is a logical impossibility so God couldn't create it by virtue of its meaninglessness rather than God's lack of power.
1) A paradox is a phenomenon which contradicts our usual experience, and therefore also our expectation and our intuition. But it is our task to find a way how to resolve the paradox. Any successful resolution improves our thinking.
An old example is Zeno's arrow-paradox. It can be resolved by the methods from calculus, see Why does Zeno's paradox seem valid but remain obviously wrong?
2) Different from a paradox is a logical contradiction, often named an antinomy. A logical contradiction consists of two statements which contradict each other. Here one has to abandon at least one of the statements.
Classical logic has no way to deal with two contradicting statements. This principle has been formalized by the law of non-contraction "not A and non-A".
I consider your example of an omnipotent being creating a stone, which he cannot lift, a logical contradiction. The contradiction is "omnipotent" versus "cannot lift".
Such examples do not prove anything about the existene or non-existence of God. Instead, they prompt us to recall the rules of logic and to improve our logical thinking.
3) I know that the example has occupied some medieval Christian theologicans.
Kant addresses this kind of notion with his Antinomies: https://en.wikipedia.org/wiki/Kant%27s_antinomies.
His conclusion is that human logic is essentially limited in at least these four ways. Russell's paradox introduces a fifth. If it cannot resolve the ultimate nature of something like the start of time, or definition of a set, it also cannot decide the issue of God. We simply have to give up.
The argument here is an antinomy like those, there is a problem with our ability to combine the notion of mass or force, which, as quantities, are implicitly limited, with the notion of perfection, which is implicitly unlimited.
This is a limitation of our intuition, which means we can't use that intuition to decide whether or not God exists. After all, God's involvement is secondary, right? You can remove him entirely and still have the problem. The question "What is the outcome of an irresistible force acting on an immovable object?" is not about God.
Your question is interesting because many people tend to draw the opposite conclusion: If a paradox is shown to exist, then this supports the idea of the existence of God. Keep in mind that they don't claim that paradoxes are in themselves knockdown proofs of God's existence, just that they are arguments in favor of the idea. Their reasoning tends to follow the below pattern:
- The existence of paradoxes means that logic and math can't prove everything.
- Somethings are true that can't be proved by logic and math, a) therefore the only other possible explanation for such things is God or b) The existence of God is one of those things that can't be proven using logic or math.
See for example the following article, by Marko Vojinovic:"Reductionism, emergence, and burden of proof" — part I and part II , and the SEP article on Godel's incompleteness theorems - section 6.5 and the reference therein.
On the other hand the physicist Mario Gleiser offers an argument related to yours in his book "A Tear at the Edge of Creation" (I will need to find my copy and then I will provide a more complete reference). His argument is physics based, not logic based, and goes something like:
The search for a unified theory of everything is useless, empirical evidence indicates that there is no such theory. (His implication here is that there will always be paradoxes that arise in the natural science, for example those that arise from the conflict of Relativity and Quantum Mechanics - note Gleiser doesn't believe that String Theory is going anywhere).
The lack of a single unified theory disproves monotheism, since the idea that one scientific theory should be able to explain all natural phenomenon is really just a form of monotheism in itself.
In conclusion, the existence of a logical paradox wouldn't really say one thing or the other about God's existence, and as Jobermark pointed out, we simply have to give up.