Is it Islamically permissible to pursue the private lives of artists who commit morally compromising acts and publicize them with the intention of exposing them?

The default is the impermissibility of seeking out the faults of Muslims, spying on them, and backbiting them, as evidenced by the Almighty's saying: "O you who have believed, avoid much [negative] assumption..." and the Hadith: "O you who have believed with your tongues...".

As for the one who openly commits sins and فسق (transgression), and seeks to spread them, there is no backbiting for him, because he has forfeited his own sanctity. He is threatened with punishment for loving that lewdness should spread, as the Almighty said: "Indeed, those who like that immorality should spread among those who have believed..."

Exposing these open transgressors is legally required, especially if people are deceived by them and take them as role models; this falls under the category of giving advice to Muslims, as Imam Ibn Rajab stated. The means of exposure must be lawful and should not involve any forbidden act or lead to a greater mischief.