It comes after Mr Musk announced he was walking away from his proposed $44bn (£37bn) takeover of Twitter on Friday.
He claimed Twitter had not given information about the number of fake and spam accounts on the platform.
Now Twitter has asked a Delaware court to order Mr Musk to complete the merger at the agreed $54.20 per Twitter share.
"Having mounted a public spectacle to put Twitter in play, and having proposed and then signed a seller-friendly merger agreement, [Mr] Musk apparently believes that he - unlike every other party subject to Delaware contract law - is free to change his mind, trash the company, disrupt its operations, destroy stockholder value, and walk away," said the lawsuit.
The lawsuit went on to accuse Mr Musk of "a long list" of violations of the merger agreement that "have cast a pall over Twitter and its business".
Twitter chairman Bret Taylor tweeted that the microblogging site wanted "to hold Elon Musk accountable to his contractual obligations".
Mr Musk tweeted on Tuesday: "Oh the irony lol [laughing out loud]."
The lawsuit said Mr Musk had backed out of the deal because it "no longer serves his personal interests".
Mr Musk is the chief executive of electric car company Tesla.
The lawsuit said that after Mr Musk agreed to the deal, the stock market fell, along with Tesla shares.
"The value of Mr Musk's stake in Tesla, the anchor of his personal wealth, has declined by more than $100bn from its November 2021 peak. So [Mr] Musk wants out," it said.
"Rather than bear the cost of the market downturn, as the merger agreement requires, [Mr] Musk wants to shift it to Twitter's stockholders," it added.
