The billionaire followed up Monday with his previously announced plans to sue Media Matters in response to a recently published story suggesting pro-Nazi advertisements were running alongside organic user content ... an assertion Elon claims is BS.

According to the lawsuit, obtained by TMZ, Media Matters "knowingly and maliciously manufactured" side-by-side images depicting advertisements next to neo-Nazi and white nationalist posts on his X app, formerly Twitter.

X claims Media Matters is driving advertisers from Elon's platform by passing off the images as something the typical X user might encounter on the app ... something Musk says is far from true.

What's more, X alleges Media Matters went to great lengths to manipulate the X algorithm to create situations where paid posts from huge companies would appear next to racist and incendiary content.

X says Media Matters then took screenshots of the posts as the basis of a negative story ... "leaving the false impression that these pairings are anything but what they actually are: manufactured, inorganic, and extraordinarily rare."

The suit says it's all part of a wide-ranging smear campaign Media Matters is mounting against Elon and X ... claiming the group is hellbent on driving advertisers away from the platform in order to destroy Elon's company.

As we reported ... Elon announced Friday plans to file a "thermonuclear" lawsuit against Media Matters for what he says was a "fraudulent attack" on his app.

There's lots of reasons Elon's pissed ... including the fact the Media Matters report seems to have spurred a mass exodus among HUGE companies advertising on X.

Media Matters is not backing down though ... its president, Angelo Carusone, tells TMZ ... "This is a frivolous lawsuit meant to bully X’s critics into silence. Media Matters stands behind its reporting and looks forward to winning in court."