Maranatha:
This depends on what you mean by proven.
Is there a scientific laboratory test that can repeatedly demonstrate the existence of God? – No, Science is limited to this universe.
Can we use logic and reason to prove God’s existence beyond a shadow of a doubt? – No, but there is almost nothing that can be proven beyond a shadow of a doubt since it’s impossible to use logic to prove the existence of reality.
Can we (and does the Summa) prove God existence beyond a reasonable doubt? - Yes, if you can look at the logic and the data objectively.
St. Thomas does not try to prove it - he makes arguments to demonstrate it. Which is not quite the same thing.
God’s existence can’t be proved, because God is the God revealed in Jesus Christ - there is no other God. God, and that God Who is revealed, are the same God - to demonstrate or prove the one, is to demonstrate or prove both. And since this revelation is revelation, & therefore depends for its being revealed solely on the gracious and free initiative of God, it is impossible to prove that God exists: for that would be tantamount to proving by human reason, what can be known only by God’s revelation. God must come down to us - we cannot climb up to God: unless He comes to us first. Which He does supremely in Christ.
The God and Father of Our Lord Jesus Christ is not (say) a deist’s God, with Christian accretions - the deist’s God is an anorexic version of the God of Jesus Christ, the God of Abraham, the God, the God of Jacob. The scholastic method is too frail to demonstrate any deity but a bundle of qualities, who is far less than the God made known by revelation. The God of Aristotle is not the God of Jesus Christ, but a weak & distorted echo of Him, and as such, very misleading.
Apart from which, it is impossible to be wholly objective - we are human beings, and sinful ones: not purely spiritual entities such as the angels. And as we are within the universe, we cannot know what - Who - exceeds the universe by looking at what is in the universe, since the universe is no greater than we are, compared to God. Our consciences are far more reliable than the rest of the universe as heralds of God, because they are part of a being made in the likeness of God. But even conscience is limited as evidence of God, though sufficient for its purpose - which does not include demonstration of the existence of God. ##