HOW does the existence of the Universe make those who do not worship God to be "without excuse" as St. Paul puts it?**
Ultimate no argument or theory is going to be acceptable to everyone. There will always be those who refuse to believe on matter what arguments are put before them.
To even try at a general base of understanding here we have to try to gain a foothold on the level in natural theology. St. Paul would have had a basic understanding of this argument in his days.
A theist believes there is a God, an atheist believes there is no God, an agnostic is unsure if there is a God or not. So where do we go?
A cosmological argument, in natural theology, is an argument which claims that the existence of God can be inferred from facts concerning causation, explanation, change, motion, contingency, dependency, or finitude with respect to the universe or some totality of objects. A cosmological argument can also sometimes be referred to as an argument from universal causation, an argument from first cause, the causal argument, or prime mover argument. Whichever term is employed, there are two basic variants of the argument, each with subtle yet important distinctions: in esse (essentiality), and in fieri (becoming).
The basic premises of all of these arguments involve the concept of causation. The conclusion of these arguments is that there exists a first cause (for whichever group of things it is being argued has a cause), subsequently deemed to be God. The history of this argument goes back to Aristotle or earlier, was developed in Neoplatonism and early Christianity and later in medieval Islamic theology during the 9th to 12th centuries, and was re-introduced to medieval Christian theology in the 13th century by Thomas Aquinas. The cosmological argument is closely related to the principle of sufficient reason as addressed by Gottfried Leibniz and Samuel Clarke, itself a modern exposition of the claim that "nothing comes from nothing" attributed to Parmenides.
Plato (c. 427–347 BC) and Aristotle (c. 384–322 BC) both posited first cause arguments, though each had certain notable caveats. In The Laws (Book X), Plato posited that all movement in the world and the Cosmos was "imparted motion". This required a "self-originated motion" to set it in motion and to maintain it. In Timaeus, Plato posited a "demiurge" of supreme wisdom and intelligence as the creator of the Cosmos.
Aristotle argued against the idea of a first cause, often confused with the idea of a "prime mover" or "unmoved mover" (πρῶτον κινοῦν ἀκίνητον or primus motor) in his Physics and Metaphysics. Aristotle argued in favor of the idea of several unmoved movers, one powering each celestial sphere, which he believed lived beyond the sphere of the fixed stars, and explained why motion in the universe (which he believed was eternal) had continued for an infinite period of time. Aristotle argued the atomist's assertion of a non-eternal universe would require a first uncaused cause – in his terminology, an efficient first cause – an idea he considered a nonsensical flaw in the reasoning of the atomists.
Like Plato, Aristotle believed in an eternal cosmos with no beginning and no end (which in turn follows Parmenides' famous statement that "nothing comes from nothing"). In what he called "first philosophy" or metaphysics, Aristotle did intend a theological correspondence between the prime mover and deity (presumably Zeus); functionally, however, he provided an explanation for the apparent motion of the "fixed stars" (now understood as the daily rotation of the Earth). According to his theses, immaterial unmoved movers are eternal unchangeable beings that constantly think about thinking, but being immaterial, they are incapable of interacting with the cosmos and have no knowledge of what transpires therein. From an "aspiration or desire", the celestial spheres, imitate that purely intellectual activity as best they can, by uniform circular motion. The unmoved movers inspiring the planetary spheres are no different in kind from the prime mover, they merely suffer a dependency of relation to the prime mover. Correspondingly, the motions of the planets are subordinate to the motion inspired by the prime mover in the sphere of fixed stars. Aristotle's natural theology admitted no creation or capriciousness from the immortal pantheon, but maintained a defense against dangerous charges of impiety.
Thomas Aquinas (c. 1225–1274) adapted and enhanced the argument he found in his reading of Aristotle, Avicenna, and Maimonides to form one of the most influential versions of the cosmological argument. His conception of First Cause was the idea that the Universe must be caused by something that is itself uncaused, which he claimed is that which we call God:
The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.
Importantly, Aquinas' Five Ways, given the second question of his Summa Theologica, are not the entirety of Aquinas' demonstration that the Christian God exists. The Five Ways form only the beginning of Aquinas' Treatise on the Divine Nature.
Argument from contingency
In the scholastic era, Aquinas formulated the "argument from contingency", following Aristotle in claiming that there must be something to explain why the Universe exists. Since the Universe could, under different circumstances, conceivably not exist (contingency), its existence must have a cause – not merely another contingent thing, but something that exists by necessity (something that must exist in order for anything else to exist). In other words, even if the Universe has always existed, it still owes its existence to an uncaused cause, Aquinas further said: "... and this we understand to be God."
Aquinas's argument from contingency allows for the possibility of a Universe that has no beginning in time. It is a form of argument from universal causation. Aquinas observed that, in nature, there were things with contingent existences. Since it is possible for such things not to exist, there must be some time at which these things did not in fact exist. Thus, according to Aquinas, there must have been a time when nothing existed. If this is so, there would exist nothing that could bring anything into existence. Contingent beings, therefore, are insufficient to account for the existence of contingent beings: there must exist a necessary being whose non-existence is an impossibility, and from which the existence of all contingent beings is ultimately derived. - Cosmological argument