Some more reading material on this topic:
http://www.jewishanswers.org/ask-the-rabbi-category/the-jewish-calendar-and-holidays/?p=1336
http://www.beingjewish.com/identity/kingdavidjew.html.
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Ruth and Naomi went to the Land of Israel. There, Ruth met Boaz, and they got married. They had a son named Oved. Oved had a son named Jesse, and Jesse had a son named David. That David later became the King David. So, King David was also descended from Gentiles. Was King David Jewish?
Some people argue that if Ruth was not Jewish, neither were her children, and therefore King David was not Jewish.
The first and obvious problem with that argument is that Ruth was not King David's mother. She was his great-grandmother. Let's assume for now that she did not convert to Judaism. If her son Oved married a Jewish woman, then his son Jesse was Jewish anyway. Even if Oved married a Gentile woman, if their son Jesse married a Jewish woman then their son David was still Jewish!
So even if Ruth did not convert, you still have to prove that King David's mother was not Jewish. So far, no one has furnished me with any proof of this at all. It does not matter at all whether Ruth converted, if King David's mother was Jewish.
But let us consider: did Ruth convert? Well, first of all, it's pretty logical to assume that she did, since the Torah explicitly forbids us to marry Gentiles who have not converted. In Deuteronomy 7:3 the Torah tells us, "Do not marry with them; your daughter you may not give to a Gentile's son, and you may not take a Gentile's daughter for your son." Why? Because, says the Torah, "For he will take your son away from Me, and they will worship the gods of others..." (ibid., verse 4). (See my article, "Judaism: Race, Religion, or Ethnicity?" for further explanation of this Commandment.)
Note, by the way, that the Torah's prohibition against marrying a Gentile applies to both a Jewish man marrying a Gentile woman, and a Jewish woman marrying a Gentile man. Both are forbidden.
So we have no reason to assume that Ruth didn't convert. Boaz, the leader of his generation, would not have married a Gentile woman who had not converted. Remember, Ruth went to Boaz because he was the next closest relative to her dead husband, and therefore it was his obligation to marry her to give her the children her dead husband never gave her (Ruth 2:20; 3:3; 3:12; and Chapter 4). But if she had not been Jewish, Boaz would not have had such an obligation at all!
However, since there are people who, for no clear reason, still insist that Ruth did not convert, I will demonstrate from the Book of Ruth itself that Ruth did convert.
In Ruth 4:11, it says:
And all the people at the gate as well as the Elders were witnesses, and they said, May Hashem let this woman who is joining your household be like Rachel and Leah, both of whom built the House of Israel, and may you do great things in Efras, and be considered significant in Bethlehem. And may your home be like the home of Peretz son of Judah and Tamar, from the offspring that Hashem will give you from this young woman.Note that the Elders of Israel were among the witnesses. They witnessed and approved of this marriage. Moreover, they obviously expected that Hashem will favor that marriage. Would they have felt that way if Ruth had not converted to Judaism? If Boaz was transgressing the Torah's Commandment against marrying a Gentile woman, would they have blessed the marriage? Would they have blessed her to become as great as the Matriarchs Rachel and Leah? It seems rather clear that Ruth converted to Judaism. Therefore, any children she had were Jewish. This is my own observation and the logic is inescapable.
Some argue that "there was no conversion process back then." I must wonder how they know this. They simply mean to say that they do not believe that the Laws of Judaism existed back then, and therefore it was okay to marry Gentile women. This entails ignoring the Torah's Commandment not to marry Gentiles.
Not only that, but the great people of Jewish history certainly did not consider it permitted to marry unconverted Gentiles from any other nation.
Let's take Moses, for example. Moses was married to an Ethiopian woman, Zipporah. Yet when the soldiers brought back Gentile women from Midian, Moses got angry at them (Numbers 31:14-15). What was the difference? The difference is that Zipporah converted to Judaism.
How do we know that Zipporah converted? Because we know that she kept the Commandments. When Moses failed to circumcise his son because he was afraid that the traveling would kill him, what did Zipporah do? "Zipporah took a (sharp) rock and cut off the foreskin of her son..." (Exodus 4:25). Evidently, Zipporah was an observant Jew.
Intermarriage is also mentioned in 1 Kings, Chapter 11:
King Solomon loved many Gentile women, such as the daughter of Pharaoh, Moabite, Amonite, Edomite, Sidonite, and Hittite women. They are Gentiles, about whom Hashem told the Children of Israel "Do not intermarry with them and do not let them intermarry with you, for they will surely influence you towards their religions." Those are the people that Solomon clung to in love. He had seven hundred queen-wives, and three hundred concubines, and these women influenced him.
When Solomon grew old, his wives influenced him towards their gods, and thus his heart was not complete in his service of Hashem, as his father David's was....
Hashem said to Solomon, Since this is the way you are, and you have not fully obeyed My covenant and My Laws that I commanded you, I shall tear part of the kingdom from you, and I will give it to one of your subjects. I shall not do this in your lifetime, for the sake of your father David. I shall tear it away from your son.We see here that the prophets considered what King Solomon did, in marrying Gentile women, to be a sin. Had they fully and properly converted, they would not have influenced him away from Hashem. (Bear in mind that the Talmud says that King Solomon never actually worshipped any idols, but since he did not stop his wives from doing so when he could have, Hashem considered it as if it were King Solomon's own sin.)
Even back then, evidently, there was a process of conversion. Thus we have amply proven that Ruth was a convert, just like Sarah, Rebecca, Leah, Rachel, Bilha, and Zilpah. And as I said above, in any case the nationality of Ruth would not have thrown into question the Jewishness of King David, who was her great-grandson.
I wish to reiterate that absolutely none of what I wrote here is my own opinion. Everything in this article I took from the Talmud and other Rabbinic Writings from over a thousand years ago. None of this is my own interpretation, and none of this did I think up on my own, unless I explicitly said so.