What is the ruling on the supplication: "O Allah, I seek refuge in You from pursuing the private matters of others, especially if they are Muslim, male or female"? And does the 'awrah (private parts) of a Muslim differ from that of a disbeliever in terms of the obligation to cover it?

This supplication is permissible because it seeks refuge from scrutinizing the private lives of Muslims, which is forbidden by Islamic law. In the Hadith: "O you who have believed with your tongues but into whose hearts faith has not entered, do not backbite Muslims, nor spy on their private lives. For whoever spies on their private lives, Allah will spy on his, and whoever Allah spies on his private life, He will expose him even in his own home." Narrated by Abu Dawud and authenticated by Al-Albani. And the Prophet, peace be upon him, said: "If you scrutinize people's private lives, you will corrupt them or almost corrupt them." Narrated by Abu Dawud and Ibn Hibban. And concealing a Muslim's faults has great virtue, for "Whoever conceals a Muslim, Allah will conceal him in this world and the Hereafter." It is also forbidden to scrutinize the private life of a non-belligerent disbeliever.