Just as we need to define who is a Jew, we also must define who is a righteous gentile since our second requirement is that we accept these people.
Talmudic Judaism answers this question with the
Noahide Laws. These laws have two problems. First, they aren't strict enough. A Nazi could easily qualify as a righteous gentile under the Noahide Laws. And second, these laws are hard to judge. Just as Biblic Judaism has chosen 5 requirements for Jews that are easy to judge, we must do the same for righteous gentiles.
The solution of Biblic Judaism is to generalize our requirements for Jews so that they can work for gentiles. So our requirements are:
1. Keep a significant religious ritual.
2. Accept righteous people of other religions.
3. No tattoos.
4. Dress modestly.
Now we can judge other religions based on this. Serious Christians tend to meet all requirements except the first. Their main ritual is going to church, but this isn't really a significant ritual. Historically, the biggest Christian ritual was actually keeping a Sunday sabbath. This was particularly true of the Puritans. So Christians who keep a Sunday Sabbath are righteous gentiles. What about Atheists. Atheism is clearly a religion, but it is a religion of selfishness so it has no rituals. So Atheists are never righteous gentiles. Muslims? Generally yes, Muslims are righteous gentiles. The five daily prayers of Islam is a significant religious ritual. Muslims are a borderline case for requirement #2, but they do accept people of other religions based on God. In this they are the same as Talmudic Judaism which has this requirement of God as its #1 Noahide Law. Muslims generally follow rules #3 and #4.
It is worth discussing why keeping a significant religious ritual is important for morality. The reason is because it shows dedication to one's religion, that one places one's religion above one's own laziness and selfishness. Because a significant ritual will be regular, it will be easy to judge of someone keeps it. And this makes it a good way of judging whether a person is likely to be moral.
Unlike Islam and Talmudic Judaism, we do not require righteous gentiles to believe in God. Members of religions like Buddhism and Shinto can certainly qualify as righteous gentiles.