Afterward all the sons of Israel came near, and he commanded them to do everything that the LORD had spoken to him on Mount Sinai. (Exodus 34:32)
The Law of Moses was given to the Jews at Mt. Sinai. I am not a Jew therefore the Law does not apply to me.
First, let's state the obvious that the commandments are God's. This is the Law of God given through Moses not something Moses made up on the spot:
You shall not make for yourself an idol, or any likeness of what is in heaven above or on the earth beneath or in the water under the earth. You shall not worship them or serve them; for I, the LORD your God, am a jealous God, visiting the iniquity of the fathers on the children, on the third and the fourth generations of those who hate Me, but showing lovingkindness to thousands, to those who love Me and keep My commandments. (Exodus 20:4-6)
Second, it should be noted that God instituted both physical and spiritual laws when He created everything.
There are physical laws in the universe like the law of gravity. If you jump off a tall building in an attempt to violate the law of gravity without any way to slow or cushion your fall, you will go splat.
There are also spiritual laws in the universe. Paul tells us that "the Law is spiritual" (Romans 7:14) and if a person commits theft, murder, or idolatry, there are spiritual consequences.
Just like the physical laws are universal, the spiritual laws are also universal. Paul makes the point that, even Gentiles (who are not part of the covenant), still sometimes do instinctively the things of the Law (Romans 2:14).
While God included His universal commandments in the covenant with Israel at Mt. Sinai, that does not mean they are only for Israel. Since Israel was going to be His chosen people, His nation of holy people, God had to make it clear that all of His universal spiritual laws applied to them.
Would a believer ever make the claim that it is permitted for them to commit injustice, murder, idolatry, fornication, theft, adultery, or any other thing that is clearly not permitted by "the Law"?
Romans 3 makes it clear that everyone is judged by the same Law:
Now we know that whatever the Law says, it speaks to those who are under the Law, so that every mouth may be closed and all the world may become accountable to God; because by the works of the Law no flesh will be justified in His sight; for through the Law comes the knowledge of sin. (Romans 3:19-20)
God desires that the whole world may become accountable to Him... through His Law.
If you would like to dig deeper, check out more answers to this objection.