"This will take place on the day when God will judge men's secrets through Jesus Christ, as my gospel declares" (Romans 2:16)
This is quite a statement! In the past few verses Paul has unequivocally proven that humans are without excuse before God. Both for Jews, who have the Law and for Gentiles who have the essence of the Law written on their hearts. Everyone has an ingrained knowledge of God and His standards, therefore we also have a knowledge of our condition apart from God.
At the end of time, when unbelievers stand before God for final judgment there will be no excuse available. Paul says that God will judge men's secrets through Jesus Christ, or as the Amplified Bible says, "God by Jesus Christ will judge men in regard to the things which they conceal (their hidden thoughts)." ( Romans 2:16 ) Even if they manage to disregard and suppress this knowledge in the present world, all men's secrets will be laid bare before Him. That knowledge and those thoughts will be the proof to condemn.
Mankind is without excuse but not without hope...
Paul does not want to condemn us, that is not the purpose of this letter. He longed for everyone to come into fellowship with the Lord. He is statically building the case for grace one brick at a time. In the first three chapters of Romans, he is tearing down old weak foundation and excuses to rebuild upon the firm foundation, the institution of Christ. To do that, he must destroy all arguments that counter the message of grace.
Paul calls it his gospel. Not because he created it but because he was obligated to it. He was compelled to preach the good news unto the ends of the earth. Paul receive his revelation of the Gospel directly from God. He was so sure of the message of Grace that he told the Galatians "the gospel I preached is not something that man made up. I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ." ( Galatians 1:11-12 )
Paul wanted everyone to know of the good news of Christ Jesus. That news was not to tell men of their sinful condition and hopelessness before God, but that God had sent His Son to be our substitute for the punishment of sin. He bore our sin, our shame, our punishment and gave us His righteousness, His holiness and His forgiveness. Paul's gospel by direct revelation of Christ Jesus declares we are forgiven and in right standing with God the Father. This is completed in us through faith in Jesus Christ alone. It in no way is dependant on our own performance, adhering to the law or because we are basically good.
Today, if you have only experience the condemnation that religion casts, my you find grace. May you understand the Gospel as Paul preached it. The almost too good to be true news of Christ Jesus. May you rest assured that at the end of time, you will not stand in judgment before God but you will accepted exactly as you are, a holy and righteous son or daughter of the most high King.