I don’t think that non-Jews should be required to follow any of the 613. By the text of the Tanakh they are only required to follow the Seven Laws of Moses.
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Catholics are required to observe many of the 613 Mitzvot:. Here are the ones which in my opinion are binding on Catholics. Some of the others, being open to interpretation could be interpreted in such a way that Catholics may be obligated to follow them But others, Catholics would not be obligated to follow them, according to our priests.
Here are some which Catholics must follow. Perhaps you could interpret them as being in some way related to the 7 Noahide Laws which you have mentioned. So this would be why Catholics would follow them from your point of view, because in the end they are related to the seven Laws of Noah. But Catholics generally, don’t go by that. Catholics go by what is called the “Natural Law” and also the ten Commandments and the Tradition of the Church.
I am going by the list as enumerated at:
jewfaq.org/613.htm
Catholics must obey and follow the following Mitzvot:
1,2,3,4,5,6,7,8,9,10,11.
However, please note for #6, that the unity of God in Catholicism is different from that of Judaism, since Catholics believe in the Trinity.
22, 25 (24 is encouraged, but not required).
26,27,28,29,30,31,32,
(36,37,38,39 are encouraged in charity)
40,52
53,54,55
[Catholics will not follow #56]
59,60,61,62
For 63, it applies generally, but would not apply to monks or priests. Also, Catholics seem to disregard this one to some extent if they use NFP, which is approved by the Church for serious reasons.
Catholics will observe: all of 82- 105, but in some cases they did allow in the past #106. I don’t agree with their allowing #106, but that is what happened before.
BTW, do Jews ever neuter their cats? If so, then they are violating #106, isn’t it?
109, 110, 111.
170,181,182,183.
Catholics should observe 203,204, and 205, I think.
Also, I like #206, and I think that Catholics should observe it. But with the explosion in marriage annulments given out by the Catholic Church recently, I am not sure that Catholics are following this one.
207,208,209.
241 must be observed, and some of the others here also.
271, 272, 274, 275, 276, 277.
278,279,280,281,282,283,284.
314, 315, 318, 330,331,332.
348 is a good one in my opinion, but I don’t think that Catholics are following it.
I don’t think that Catholic women follow #366, since I see them wearing pants all the time.
Also, it doesn’t look like the USA follows #600. Does Israel always follow #600? I am not sure if the Palestinians would agree that Israel follows #600, but I could be wrong on that.
Thank you for this topic.