Believe it or not, we would actually like for non-Jews to use Jewish prayers. Many Jewish prayers have universal appeal and universal themes, anything from health and wealth to finding lost objects. Yes they mention the G-d of Israel, but everyone is supposed to believe in the G-d of Israel! The book of Tehillim (Psalms) specifically mentions on a number of occasions the 'Fearers of G-d', who are non-Jews who worship G-d. Incidentally, Tehillim is a great place for non-Jews to find Jewish prayers they can use. Two other good sources of prayers for a multitude of situations are Aneni, published by Feldheim, and Likutei Tefillot, which is a two volume set of prayers written by Rabbi Natan of Nemirov (Breslov Research Institute).
However, one has keep in mind what the purpose of prayer is. Prayer is not about being or doing spiritual. Prayer is speaking to G-d. So when speaking to G-d, one should be speaking the truth. If a non-Jew were to use a prayer that represents themselves as Jewish, when they are not, it will not be a communication with G-d that is based in truth. If a non-Jew were to pray the Amida (standing prayer) for example, they would be speaking falsehood when they say the first of the 19 blessings "Blessed are you Hashem our G-d and the G-d of our forefathers, G-d of Abraham, G-d of Isaac, and G-d of Jacob." The well-meaning non-Jew may have accepted Hashem as G-d, but their forefathers most likely did not. Even if their forefathers might have, their forefathers are not Abraham, Isaac, and Jacob. For this reason, the Rabbis debated whether or not converts could say these types of blessings, with many ruling in the negative.
I know what you might be thinking. The non-Jew might feel spiritually attached to Israel, as if spiritually grafted on to the family tree of our forefathers. While such a logic has been extended to converts for the purpose of saying blessings such as the Amidah, it's never been extended beyond that and only then as not to exclude them from the community. A person is only considered a descendent of the Forefathers if they are physically descended, as G-d promised in the book of Genesis that his seed (zera) who inherit him. Even by converts their union to the Jewish people isn't complete until they marry in. One cannot just resolve to be a descendent of Abraham just like someone can't just consider themselves adopted by Bill Gates so they could get a piece of the inheritance. To use prayers that would reflect such a sentiment wouldn't be okay and actually at some level self-defeating.
My mother was an English teacher, and I am afraid that reading this question immediately raised in my mind a distinction she taught me between "can" and "may." Whether a non-Jew can read those prayers, especially in Hebrew, depends on that person's knowledge of the language and of the prayers. I presume, though, that the questioner intended to ask whether a non-Jew may read those prayers without insulting Jews or violating some aspect of the Jewish tradition. The answer to that question depends, I think, on the purpose that the non-Jew has in mind. If that person is studying these prayers as part of a process of converting to Judaism, then surely it is permissible -- even desirable -- for him or her to learn these prayers as part of preparing to do that. If the non-Jew is not thinking of converting but simply wants to learn more about Judaism, I find nothing wrong with that, unless his or her ultimate goal is to learn more about Judaism in order to convert Jews to another religion. If the non-Jew wants to use these prayers simply because they express what he or she believes, then I find no reason to object to that, but I do wonder what he or she finds meaningful in speaking about specifically Jewish practices like the fringes described in the third paragraph of the Shema and the Jewish Messianic hope articulated as part of the Amidah. Maybe this person is on the way to converting to Judaism after all -- in which case, welcome!
I suppose that a non-Jew can recite the Shema and the Amidah “sincerely”; I certainly am in no position to tell another person what she or he really believes. I would point out, though, that these are, as you put it, “specifically Jewish prayers.” More than that: they are declarations of identity. By reciting them, we Jews proclaim that we are members of a historical community that lives in covenant with God. Thus, when I recite the Shema, I am doing more than simply reading some lines from the book of Deuteronomy. I am placing myself within the community that Moses addresses in that text with the words “Hear O Israel.” Similarly, when I recite the Amidah, I begin with praise for the “God of our ancestors,” Abraham, Isaac, and Jacob (and we Reform Jews include the matriarchs Sarah, Rebecca, Rachel, and Leah in that list). In other words, I’m saying that I descend from those people, that I am a member of the people of Israel. The Amidah is not only my prayer but the prayer of all Israel as it stands before God. This accounts for the theme of “Israel” that runs throughout the text.
In other words, I have trouble understanding how a person can truly believe what these prayers state without being a Jew.
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