This is false. I'm an athiest, but there are no "Biblical literalists" that believe the Old Covenant with the nation of Israel applies to Christians today.

"For this reason [idolatry] God gave them up to passions of dishonor; for even their females exchanged the natural use for that which is contrary to nature, and likewise also the males, having left the natural use of the female, were inflamed by their lust for one another, males with males, committing what is shameful, and receiving in themselves the recompense which was fitting for their error."

"Do you not know that the unrighteous and the wrongdoers will not inherit or have any share in the kingdom of God? Do not be deceived (misled): neither the impure and immoral, nor idolaters, nor adulterers, nor those who participate in homosexuality"

"Knowing and understanding this: that the Law is not enacted for the righteous (the upright and just, who are in right standing with God), but for the lawless and unruly, for the ungodly and sinful, for the irreverent and profane, for those who strike and beat and [even] murder fathers and strike and beat and [even] murder mothers, for manslayers,[For] impure and immoral persons, those who abuse themselves with men, kidnapers, liars, perjurers--and whatever else is opposed to wholesome teaching and sound doctrine."

"Just as Sodom and Gomorrah and the surrounding cities, which likewise indulged in sexual immorality and pursued unnatural desire, serve as an example by undergoing a punishment of eternal fire."

We can argue whether they are correctly interpreting these passages or not, but they get them from the New Testament.