Quoting from our (newly upated) web page which addresses the main point of this question, which is at:
https://asknoah.org/essay/ninth-of-av
QUESTION: Is it OK for a Noahide to fast on Tisha B'Av? [The 9th of the Hebrew month of Menachem Av, when Jews observe total fasting for about 24 hours and 40 minutes, as part of their traditional mourning on this anniversary of the destruction of both the first and the second Holy Temples in Jerusalem.]
ANSWER: It would seem that if a Noahide would make a full observance of all the Jewish precepts of Tisha B'Av [especially if this includes specifically refraining from wearing leather shoes, washing for pleasure and having marital relations], he would be making a religiously-observed memorial day for himself, which is like innovating a religious observance, which is forbidden.
Rabbi Moshe Weiner, author of "Sefer Sheva Mitzvot HaShem," says that the only point upon which an individual Noahide could justify fasting is that he is mourning the temporary (but far too long) destruction of the Holy Temple and the exile of the Divine Presence. Since this is a permitted activity, it depends on his intention.
However, in this final generation of spiritual exile (which will please G-d be the first generation of the Redemption), people are not nearly as physically strong as they used to be. Therefore it is recommended that fasting should be minimized, if done at all, for those who are not obligated. Certainly a Noahide who has a medical problem should not fast if there is any health risk involved, or he should consult with his doctor as to what he can safely do.
Furthermore, the continuous obligation of Noahides is "yishuv olom," improving the condition of the world, and one who is weakened by fasting might fall short in what he could have accomplished that day for yishuv olom, G-d forbid. Also, if fasting causes one to become irritable, short-tempered or prideful, G-d forbid, there could be an actual bad effect from the fasting, which is the opposite of what needs to be accomplished. In our days it is very acceptable to substitute non-obligatory fasting with donations to proper charities (the value of at least one normal meal is suggested). Recommended charities are those which are consistent with Torah-true principles and yishuv olom. Also, the main practical intention is not for fasting, but for repentance.
Certainly a Noahide is encouraged to pray that the Third Holy Temple shall be established by Moshiach ben David very speedily in our days. And it very appropriate for a Noahide to read the Book of Lamentations on the night and/or day of Tisha B'Av.