While Paul wrote Galatians primarily to Gentiles, they were Gentiles that Judaizers were telling them that they also needed to follow the Law of Moses and be circumcized.
The major theme is Galatians is:
16 yet we know that a person is not justified by works of the law but through faith in Jesus Christ, so we also have believed in Christ Jesus, in order to be justified by faith in Christ and not by works of the law, because by works of the law no one will be justified.
(Gal. 2:16, ESV)
Paul's reference to circumcision obviously points to the Law of Moses.
Look: I, Paul, say to you that if you accept circumcision, Christ will be of no advantage to you. 3 I testify again to every man who accepts circumcision that he is obligated to keep the whole law. 4 You are severed from Christ, you who would be justified by the law; you have fallen away from grace.
(Gal. 5:2–4, ESV)
Paul pointed out that Jesus was born a Jew under the Law of Moses to fulfill the Law and become our righteousness that we could not fulfill on our own.
I do not nullify the grace of God, for if righteousness were through the law, then Christ died for no purpose.
(Gal. 2:21, ESV)
And because of him you are in Christ Jesus, who became to us wisdom from God, righteousness and sanctification and redemption,
(1 Cor. 1:30, ESV)
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.
(2 Co 5:21, ESV)
Thus, in Gal. 4:4 Paul referred to the Law of Moses.
But when the fullness of time had come, God sent forth his Son, born of woman, born under the law, 5 to redeem those who were under the law, so that we might receive adoption as sons.
(Gal. 4:4–5, ESV)
Paul wrote that the Gentiles are under the Law of Moses as far as what it written in their hearts.
For when Gentiles, who do not have the law [of Moses], by nature do what the law requires, they are a law to themselves, even though they do not have the law. 15 They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them 16 on that day when, according to my gospel, God judges the secrets of men by Christ Jesus.
(Rom. 2:14–16, ESV)
Example of the Law of Moses written in one's heart:
For the whole law is fulfilled in one word: “You shall love your neighbor as yourself.”
(Gal. 5:14, ESV)
A renowned Greek professor didn't mention the lack of the article in the Greek, but wrote:
Born under the law (γενομενον ὑπο νομον [genomenon hupo nomon]). He not only became a man, but a Jew.
Robertson, A. T. (1933). Word Pictures in the New Testament (Ga 4:4). Nashville, TN: Broadman Press.
Note the following verses also have ὑπὸ νόμον without the article:
Rom. 6:14,15; 1 Cor. 9:20; Gal. 3:23; Gal. 4:22; 5:18. Only Paul used ὑπὸ νόμον in the New Testament and always without the article. It was not used in the Septuagint.
But, Gal 3:24 has he article:
Now before faith came, we were held captive under the law, imprisoned until the coming faith would be revealed. 24 So then, the law was our guardian until Christ came, in order that we might be justified by faith.
(Gal. 3:23–24, ESV)