Can a non-Jew pray and take part at temple mount western wall prayers in Jerusalem?
The short answer is yes, non-Jews may enter the Western wall plaza and pray because it is not part of the temple mount sacred area. The restrictions according to halacha involve areas of the temple mount itself.
The western wall is not actually part of the temple area. There are areas on the temple mount where even Jews who are tamei meis may go and areas that are forbidden. During the time of the temple, there were walls set up to show where these areas were.
From the chayl,19 gentiles, those who contracted impurity from a human corpse, and those who had relations with a niddah20 are sent away.21 A person who immersed that day may enter there, for he has already immersed.22
19. The rampart surrounding the walls of the courtyard (see Hilchot Beit HaBechirah 5:3).
We actually have a Herodian stone containing the warning which showed where this line was drawn.
Carved in bold Greek letters, 2,000-year-old Herodian inscription marked off the section of Jerusalem’s most sacred site where gentiles couldn’t go — and shows they were welcome elsewhere in holy area
There are ten levels of sanctity in the Land of Israel, eight of them within the Temple Mount. As the level of sanctity ascends, those permitted to enter decrease. A gentile, for example, was not permitted to enter even the “Chail”- which was on the outer perimeter of the “Ezrat Nashim” (women’s court). Jews who were not Cohanim could enter the “Ezrat Cohanim” (the court of the Cohanim) even if they were free of all defilement (“tumah”), only to play their role in the offering of their sacrifice (Hilchot Beit HaBechira 7,19).
15 The Temple Mount is holier than [the city of Jerusalem]. Zavim, Zavot, Niddot, and women who have given birth may not enter there. [However,] a corpse may be brought into the Temple Mount and one has contracted ritual impurity from a corpse may definitely enter there.
16 The chayl is holier than the Temple Mount. Gentiles and those who contracted impurity through contact with a corpse or engaging in sexual relations with a Niddah may not enter there.