Should two idolaters come before you to have their dispute judged according to Jewish law. If they both desire to be judged according to Torah law, they should be judged accordingly. If one desires to be judged according to Torah law and the other does not, they are only forced to be judged according to their own laws.
If there is a dispute between a Jew and an idolater: If the Jew will fare better according to their laws, they are judged according to their laws. When the judgement is rendered, the judges explain: 'Your law obligates this judgement.' If the Jew will fare better according to our laws, they are judged according to Torah law. When the judgement is rendered, the judges explain: 'Our law obligates this judgement.' It appears to me that this approach is not followed in regard to a resident alien. Rather, he is always judged according to their laws.
Similarly, it appears to me that in regard to respect and honor and also, in regard to charity, a resident alien is to be treated as a Jew for behold, we are commanded to sustain them as Deuteronomy 14:21 states: 'You may not eat any animal that has not been properly slaughtered... give it to the resident alien in your gates that he may eat it.' Though our Sages counseled against repeating a greeting to them, that statement applies to idolaters and not resident aliens.
However, our Sages commanded us to visit the gentiles when ill, to bury their dead in addition to the Jewish dead, and support their poor in addition to the Jewish poor for the sake of peace. Behold, Psalms 145:9 states: 'God is good to all and His mercies extend over all His works' and Proverbs 3:17 states: 'The Torah's ways are pleasant ways and all its paths are peace.'