It is incorrect to say this. Science is most certainly capable of proving the existence of God, and not just any God, but the Christian God, and it is profitable to pursue this approach of understanding God.
Why is this so? If a person of finite ability and means were to construct a universe of moderate subtlety, or a random act of a dead universe, or if God were to create such a limited universe that falls far short of what He is capable, then you could not distinguish the results. You could not tell one such universe from another. But the Bible does not declare that we live in such a universe.
The heavens declare the glory of God;
the skies proclaim the work of his hands. (Psalm 19:1)
If God is infinite and glorious, then His infinite subtlety, majesty, attributes of love and justice and all the other characteristics that make God God must be present in His creation. Of course, being finite beings, we cannot discover the full pattern of what God created. However, we should find there is no end to the mysteries. Our science should never reach the end of the matter.
In chapter 28 of Job, the suffering man laments that though he search every part of the creation, he can never find wisdom; only God knows the way to it. After speaking of scientific attempts to search out wisdom though human effort, such as digging in the earth for precious gems, Job concludes with this:
Where then does wisdom come from?
Where does understanding dwell?
21 It is hidden from the eyes of every living thing,
concealed even from the birds in the sky.
22 Destruction and Death say,
“Only a rumor of it has reached our ears.”
23 God understands the way to it
and he alone knows where it dwells,
24 for he views the ends of the earth
and sees everything under the heavens. (Job 28:20-24)
This would seem to back up your statement that man cannot find this ultimate wisdom (which is true), except for one observation. In God's speech to Job from the whirlwind (Job chapters 38-41), He argues that every facet of the world, from the stars in the heavens, to the birds in the sky, to the earth and its creatures, to the grave and the forces of evil behind it all testify about His wisdom and reveal it. We cannot ascend to Heaven on our own effort, but in Christ, God has descended to earth to give us the Holy Spirit, who shows us the path to wisdom. The Spirit-filled woman or man can, by observing and reasoning about nature (science), retrace the path to God's wisdom which formerly was impossible to follow.
The inability to find wisdom in nature was Job's despair (and humility) speaking. The ability to find wisdom in nature was God's gracious gift, but it requires an infinitely capable interpreter - the Holy Spirit - in order to deliver accurate truths.
9 However, as it is written:
“What no eye has seen,
what no ear has heard,
and what no human mind has conceived”—
the things God has prepared for those who love him—
10 these are the things God has revealed to us by his Spirit.
The Spirit searches all things, even the deep things of God.
11 For who knows a person’s thoughts except their own spirit within them?
In the same way no one knows the thoughts of God except the Spirit of God.
(1 Corinthians 2:9-11)
One writer used an analogy that I like: ink on paper. The ink and paper represent nature. They are material and cannot truly show the wisdom of God. However, if God takes that ink into His pen and writes with that ink upon the paper that is the world, then his existence and intelligence and purpose can be clearly manifest. The ink has not changed. The paper has not changed. The only difference is that that ink is now formed into words. Those words carry information that was not there before. That information is not a material property of the ink and the paper, it is an immaterial property. God's wisdom is such an immaterial property and it is written upon our world. And as God told Job, it can be understood.
It is the glory of God to conceal a matter;
to search out a matter is the glory of kings. (Proverbs 25:2)
God would not urge us to search out what He has concealed if it were futile.