This seems hard for those who base their world-view exclusively on science to understand, that for most of them (except for those who have done the experiments themselves), their belief in the claims of science requires about the same kind of faith as those who believe in God's word does. Propaganda of the world has done of good job of fashioning such scientific claims in a positive light, making those who counter these "evidences" look like fools. However to make the statement that the discoveries come through scientific experiments are truths ignores the problem of bias which is unavoidable in every experiment even if it just the bias the researcher brings to it by being limited by their own perspective. It is probable also that many scientists do understand the uncertainty of their findings and thus they need to, or should, acknowledge the faith that makes up their adherence to the results of their experiment at least if they wish to avoid hypocrisy and double-standards.
I have been surprised by the number and vehemence of naturalistic atheist denials of their own bias on the question of theism even though once upon a time I was there myself! For what it might be worth, post-1947 (Everson v Board of Education) in the US, wall of separation doctrine as it subsequently developed seems to place secular naturalism in a kind of moral high ground in arbitrating religious claims. While independence from partisan conflict may afford some hedging against bias in the case, claims of no bias in the case of theism is a myth, as I attempt to substantiate below.
Likewise, Christians have no scientific grounds upon which to base their beliefs in the God of the Bible--we walk by faith and not by sight. While there are many evidences to us of God those are subjective to us. To make himself known, God had to reveal himself to us; therefore there will never in this world at least, lest God makes it possible, be any empirical evidence of him. The closest thing to evidence of God, the kind which the researcher might accept, is the logical one the apologetic will give of how there is logical evidence for a God (you'll have to ask the apologist for explanation of that, I'm not one). Yet, that does not give evidence of the Christian God.
I'm unclear quite what you mean in some respects. If empirical naturalistic evidence (I'm
not talking about quotes from Tacitus or Ireneus or the like) has been found to confirm the Trinity or the "Jesus = OT Messiah" equation (I say "OT" because of the thematic and conceptual baggage developed for the term "Messiah"), I am unaware of it even having been vetted (Trinitarian rocks?), or perhaps such evidence (The Shroud of Turin?) might be read as evidence for the Trinity or Jesus only for a minority. And I agree that for regeneration and conversion, God has to break through our willfully darkened and divinely hardened minds and hearts and reveal Himself to us in a converting way.
I would, however, argue that God has revealed something about His divine power and Godhead in His creation, and that to this extent God is "clearly perceived ... in the things that have been made" (Rom. 1:20). "What can be known about God [apparently including His wrath] is plain to them because God has shown it to them" (v. 19). Yes, what follows is ingratitude, human suppression of such revealed truth, and "a debased mind" (v. 28, but of course see the whole context). Thus when you deny there is "any empirical evidence of [God]," you seem also to deny some of Paul's argument in Rom. 1. If this is your intent, I demur. Empirical evidence for God is ubiquitous in the creation even if with our five senses we cannot directly experience the Spirit-God.
Furthermore you claim, "Christians have no scientific grounds upon which to base their beliefs in the God of the Bible." Given my argument in the above paragraph, it is difficult for me to know quite what you mean. Do you mean there is no scientific grounds for the existence of Jerusalem in David's or Jesus' day? No archaeological grounds supporting various biblical narratives? Now granted, there is a gap between (1) seeing scientific grounds for the fall of the walls of Jericho circa Joshua's day and (2) "belief in the God of the Bible," but for those who believe in the God of the Bible, does the scientific grounds behind the conclusion that the walls of Jericho fell circa Joshua's day necessarily comprise no basis for that belief in any terms whatsoever?
Can there be no evidence bolstering or suggesting faith in some object? If Christians drive cars by faith and not by sight, are they obliged to close their eyes while driving? Can Christians who drive cars with their eyes open not also exercise faith in driving? And if there is a "subjective" element to driving cars (we may not see the road as it really is at times), is there no objective truth to be had in the driving (or in exercising the Christian religion)?
Empirical evidence, I am arguing, sometimes places us under moral obligations that the evidence only suggests. (You see a leaking crack in an earthen dam. Tell others to run even though you do not know the dam will collapse.) If God reveals Himself to some to the saving of their souls, shedding light upon darkened minds through the gospel, does He never use means in doing so? Means which may include, though they can never be limited to, empirical or scientific evidence of some sort in the realm of the created order? True, God revealed Himself to me in the gospel of His Son according to the Bible, but along the way He also used empirical and scientific evidence, for example to cause me to doubt my atheism or to understand what a father is.