Federally, not since 1981. On the state level, some states do provide for a process that is recognizably analogous but still differ state to state, sometimes greatly. Also, states differ as to how direct a route a citizen-initiated complaint can directly affect the actions of the judicial system - some states use prosecutors as a gatekeeping mechanism, while others allow it as a process distinct from the prosecutorial system and so would skip over the gatekeeper and directly go to the judge. I don't know about most states, but I know that it's a mechanism that New Jersey, Washington, New Hampshire, and Idaho recognizes.

This is a sufficiently niche field of even criminal law that would require you to hire a lawyer to handle anyway. And usually there's some statutory limitation on what sort of prosecution a private individual can initiate. It's generally just not worth the trouble.

The DMCA does allow for a separate, federal, and civil cause of action to recover damages if the filer of the takedown notice knowingly and materially misrepresented the operative parts of the notice. 17 U.S.C. § 512(f) is where this cause of action resides. However, to prove "knowingly" and "materially" at the same time is a pretty high bar, although it might be possible to get a default judgment if they simply don't respond. Still, this is a limited remedy that most people won't be able to take advantage of since it involves potentially costly litigation and no punitive damages. It's not a prosecution of any sort, but it's sadly the best (only) option on the books that counts as relief, however limited it is.