Answering Amen to the blessing of Gentiles that are not idolaters:[1]

If a gentile[2] recited a blessing amongst one the blessings instituted by the Sages[3] one is to answer Amen if he heard the entire blessing from his mouth.[4] If however one did not hear the entire blessing from his mouth he may not answer Amen. Kutim have the same status as gentiles in this regard.

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[1] Admur 215:2; Rama 215/2; Rabbeinu Yonah Brachos 51b; Yerushalmi

Other opinions: Some Poskim rule one is never to answer Amen after the blessing of a gentile. [Kesef Mishneh in opinion of Rambam; Bach 215; brought in M”A 215/3; Mateh Yosef, P”M 215 A”A 3 and Kaf Hachaim 215/14 that so is the ruling Michaber who omitted the ruling of the Rama] However, one may answer Amen after a Yishmaeili, being they do not serve idolatry. [Maharikash; Kaf Hachaim 215/15]

[2] The wording in Admur ibid “Nachri”, however the Rama ibid writes “idol worshiper”. The M”B 215:12 explains that a common idol worshiper does not intend towards idolatry when saying Hashem’s name and hence one is to answer Amen to his blessing. It is possible to learn this way in Admur as well, as Admur only forbids answering Amen to a person who is fervently attached to idolatry, and hence perhaps your normal gentile would be valid.

[3] If the blessing was not instituted by the Sages: If a gentile states may our G-d help you or save you one should not answer Amen to his blessing. If however he says May your G-d help you or save you then one may answer Amen. [Sefer Chassidim 427; Chesed Lealafim 215/4; Betzeil Hachochmah 3/39]

[4] Obligation or voluntary: Some Poskim rule one is not obligated to answer Amen for a blessing said by a gentile and it is merely voluntary to do so. [Taz 215/3; P”M 215 M”Z 2; M”B 215/12; Kaf Hachaim 215/14] Admur however omits this opinion and implies that it is obligatory. Vetzaruch Iyun