I most certainly do deny that the Gentile is "under sin" in relation to the Law of Moses. Many Gentiles were born, lived, and died without ever hearing about the existence of the Law of Moses.
Yes the "it is written" bit tells us that the Gentile is also under sin - so I agree with that. But this does not mean that the Gentile is under sin specifically in relation to the Law of Moses. He is, as you seem to agree, under sin in virtue of "natural law".
Suppose you believed that God has revealed the content of the US Constitution to the American people. Even if the content of the US Constitution is "informed" by the same general moral principles that all human being are given in the form of "a law of conscience", this does not mean that non-Americans are subject to the US Constitution. They obviously are not.
I do not deny this "connection" you speak of. But the existence of the connection does not mean the Gentile is under the Law of Moses.
The bottom line is this: The fact that the Gentile is "under sin" and the fact that the Jewish scriptures say that the Gentile is under sin does not mean that he is under the Law of Moses.
So the "law" in 3:19 can still be the Law of Moses as applicable only to Jews.
Your response are irrational for several reasons. First, if the U.S. Consititution was written by God instead of by men and it was written instilling within it principles that are UNIVERSAL LAWS then whether you were a US citizen or not you still are in violation of the same UNIVERSAL PRINICPLES whether they are in or outside the US constitution by the fact that the Author is not limited to any race and the UNIVERSAL laws by their very nature are not limited to any race. The irrational aspect of your argument is your failure to carry through consistently the proper analogy and recognize that it was not men or the US that is the author but God and therefore UNIVERSAL LAW written into the US constitution is just as applicable to those outside the US as those inside becuause those particular laws are UNIVERSAL by their very nature and that the Author cannot be limited to the US.
Likewise, the law of conscience and nature are UNIVERSAL and their inclusion into the Mosaic Law does not make them less Universal and therefore less applicable than universal. Just because a universal law is found in an ethnic constitution does not invalidate its UNIVERSAL nature or application. The moral laws of God found in Jewish law does not make them less "moral" or less applicable to all God's creatures especially when the same law is written upon their conscience. Hence, violation of those UNIVERSAL laws are violations of them wherever they are found.
Secondly, Romans 3:10-18 does not say Gentiles are sinners per se. It is UNIVERSAL language that includes all mankind due to HUMAN NATURE that is universal and cannot be confined to any one ethnic group.
Thirdly, Romans 3:19-20 continues to use UNIVERSAL terms that cannot be restricted to any ethnic group. The law in Romans 3:19-20 is GOD'S LAW concerning GOD'S RIGHTEOUSNESS and it includes ALL universal aspects of that law as law of conscience and nature as well as all humanity under those universal aspect in addition to the Jews and those aspects relating to the Jews. God's moral laws are universal by nature and cannot be restricted to any ethnic race of mankind.
Fourthly, Romans 3:9-18 has no conclusion without Romans 3:19-20 and you cannot eradicate the Gentiles from Romans 3:9-18 any more than you can eradicate the Gentiles from the conclusion of Romans 3:19-20. Absolute proof is that Paul does not change the UNIVERSAL applications in Romans 3:9-18 to ethnic terms in Romans 3:19-20 but continues to use UNIVERSAL APPLICATION TERMS that transcends race, gender or status among mankind.
1. "EVERY mouth" not "jewish mouth"
2. "ALL the world" not "Israel" or "Jews"
3. "NO FLESH" not "Jewish flesh"