False.
Without God, men are incapable of seeing the light, so if God turns away then men WILL believe what is false.
The Bible also says that God inscribed his laws in our hearts. It's like saying God put money in our pockets. What we do with it, is our decision—to our benefit or perdition.
Read Hebrews 8. The end says (never mind that this only pertains to the Jews), with my emphases:
11 No longer will a man teach his neighbor, nor a man his brother, saying, 'Know the Lord,' because they will all know me from the least of them to the greatest.
12 For I will forgive their wickedness and will remember their sins no more. [Heb 8:10-12]
It tells us (verse 10) that there will be NO NEED for PREACHING because men will KNOW God in their MINDS and will have His laws in their HEARTS.
What they do after that will be either their blessing (obeying the Lord) or their downfall (abusing the blesisngs He gave), because He has forgiven the sins of ALL.
What the Bible says is the same as knowing the Ten Commandments, but without having the ability to carry them out perfectly, on our own. It is enough to avoid excuse, but it is not enough for salvation. Salvation comes by grace only given to some.
Read Hebrews 8. The end says (never mind that this only pertains to the Jews), with my emphases: ...
The whole of verse 10, which you omit, says that the Covenant is with the "House of Israel". Paul shows us who the House of Israel is, all believers, not Jews by birth.
What they do after that will be either their blessing (obeying the Lord) or their downfall (abusing the blessings He gave), because He has forgiven the sins of ALL.
By Christ's work He has forgiven all the sins of all believers, not all people. Otherwise, you would have salvation being completely determined by the actions of man. I don't recall your ever saying that before.